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# David Finkelstein

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### Preferred citation

In footnotes or endnotes please cite AIP interviews like this:

Interview of David Finkelstein by Dean Rickles on 2013 February 18,

Niels Bohr Library & Archives, American Institute of Physics,

College Park, MD USA,

www.aip.org/history-programs/niels-bohr-library/oral-histories/40665

For multiple citations, "AIP" is the preferred abbreviation for the location.

### Abstract

In this interview David Finkelstein discusses topics such as: his childhood; undergraduate work at City College in New York in electrical engineering and physics; Gabriel Kron; John von Neumann; Mark Zemansky; graduate work at Massachusetts Institute of Technology (MIT); general relativity; Victor Weisskopf; Norbert Wiener; Stevens Institute of Technology; moving to an Atomic Energy Commission (AEC) laboratory at New York University during the Korean War; relativity; Charles Misner and working with him; Richard Feynman's work; Dennis Sciama and Roger Penrose; European Centre for Nuclear Research (CERN); Yeshiva University; Martin Kruskal; space-time; quantum mechanics; David Bohm; and Yakir Aharonov.

### Rickles:

The date is February 18, 2013. Our location is Professor David Finkelstein’s home in Atlanta. So it’s usual to begin these kinds of oral history interviews by looking at early family life and general family background and what your parents did and this kind of thing. I read your CV online. I know that you were born July 19, 1929 in New York City. So into what kind of family was that?

### Finkelstein:

Both my parents came from the same small town in Poland not far from Warsaw. It’s called Sokolow. That’s Polish for small town.

### Rickles:

When did they come over to America?

### Finkelstein:

Around 1914 or 1915.

### Rickles:

Did they bring any skills with them? Were they educated or have trades?

### Finkelstein:

My father was ordained as a Rabbi before he left Poland, but he didn’t — He was offered a seat in New York when he arrived, but he decided it wasn’t for him and went into the grocery business.

### Rickles:

Whereabouts did they live in New York?

### Finkelstein:

My babyhood was in Brooklyn.

### Rickles:

Right. So did they have any kind of scientific interests or mathematical interests at all? Or have any interest in learning about science?

### Finkelstein:

My father, no. My mother read a little in behaviorism and studied languages, but not science, no.

### Rickles:

Do you have any brothers or sisters?

### Finkelstein:

No.

### Rickles:

So you’re an only child. So you lived in Brooklyn. So your first school was in Brooklyn.

### Finkelstein:

Public schools.

### Rickles:

Which school was that?

### Finkelstein:

The earliest one I remember was near Flatbush Avenue. I don’t even remember the number of it any longer.

### Rickles:

Were there any people at that school who went later on to — was it Stuyvesant?

### Finkelstein:

Pardon?

### Rickles:

Were there any people from that early school or from where you lived who continued on with you to later life that you were close to?

### Finkelstein:

No. My parents moved rather often as they advanced from one store to another, and that always involved a completely new set of friends.

### Rickles:

Ah, okay. Did you display anything? So what age are we talking? In Primary school?

### Finkelstein:

I migrated to New York City, Manhattan, at about the age of 12.

### Rickles:

Okay, and then you went into Stuyvesant.

### Finkelstein:

First I went to Joan of Arc Junior High School, and then to Stuyvesant.

### Rickles:

All right. So before you went to Stuyvesant, had you displayed any mathematical, physics-based kind of skills? Did you show —

### Finkelstein:

I was kind of the head of my class in mathematics and behind in language.

### Rickles:

Do you remember any particular books that got you interested and excited?

### Finkelstein:

I was asking my algebra teacher questions about calculus while I was at Joan of Arc. I suppose I must have gotten my hands on Silvanus P. Thompson, Calculus Made Easy because that’s where I learned calculus from.

### Rickles:

Yeah. That’s the famous Feynman one, isn’t it? “What one fool can do, any fool can do.”

### Finkelstein:

Ancient Simian proverb.

### Rickles:

Yeah! [Laughter] Excellent. So you mentioned that particular teacher. He must have —

### Finkelstein:

Mrs. Reinhart.

### Rickles:

Right, so must have taken an interest. Was there a bit of an influence there?

### Finkelstein:

Well, she was just a lively woman. I don’t think she paid any particular attention to me.

### Rickles:

Then on to Stuyvesant High School. Was that competitive back then? It’s kind of competitive to get in there, so you have to —

### Finkelstein:

There was an entrance exam, and they skimmed. I’m very grateful to the New York public school system.

### Rickles:

What date are you talking for your entrance there? Do you remember? Do you know?

### Finkelstein:

1946.

### Rickles:

To enter Stuyvesant. Is that right?

### Finkelstein:

I spent three years there, ‘46 through ‘49. Wait. No, no. I’m sorry. That was City College, ‘46 through ‘49. So Stuyvesant must have been like ‘42 to ‘45.

### Rickles:

So right obviously slap bang in the war years.

### Finkelstein:

That’s right.

### Rickles:

So do you remember that happening? Do you remember what happened to the teaching staff? Did anything switch around?

### Finkelstein:

I distinctly remember the Sunday when the headlines said war. The Japanese had attacked Pearl Harbor. That was when I was still living in Brooklyn.

### Rickles:

Yeah, that was December 1941. Did you have as a teacher Melba Phillips, by any chance? She was there, right? Was she there?

### Finkelstein:

I didn’t know that. Where? Stuyvesant or —?

### Rickles:

Maybe it was City College of New York. I’m not sure which one. I think it was Stachel who… I’ll need to check that. I know she was at one of those. So I read somewhere that you were friends with Jacob Schwartz quite early on.

### Finkelstein:

Right. Quite close friends. I think of him as my best friend. We went through high school and City College together.

### Rickles:

Okay. So what kind of interactions did you have? Were you challenging each other with math problems? Were you reading books together or teaching each other?

### Finkelstein:

We talked in high school, and in City College, we often took — not often — once or twice we took honors courses together. We would have study sessions. Jack had the theory that you could get by with the least sleep if you broke your sleep up after a couple of hours. So he would sleep two hours, read as long as he could, sleep two hours.

### Rickles:

I tried that.

### Finkelstein:

I went through Weil’s Algebraic Geometry that way.

### Rickles:

While still at high school?

### Finkelstein:

No, that was in college.

### Rickles:

Right, this is in college. But you were friends with him in high school as well, right? So do you remember what kinds of things you were discussing in high school, what kind of topics or what was getting you excited?

### Finkelstein:

I dredged up the memories for the In Memoriam that I contributed to his volume. I distinctly remember riding home with him in the subway and asking him the difference between mathematics and physics. At that stage, from all I knew, they were both fiddling with equations.

### Rickles:

So what kind of age would that have been, by the way, roughly? Still high school.

### Finkelstein:

That would have been probably… Since he entered Stuyvesant I believe a year after me (he was younger than I was), that would probably be my second year at Stuyvesant.

### Rickles:

Did you discuss things like logic and computing? I know computing was at a very —

### Finkelstein:

Not in high school, but very much in college. He was really influenced by his studies under Emil Leon Post.

### Rickles:

Yeah. So Post was at City College as well, right. Yes. John Stachel was —

### Finkelstein:

As was Martin Davis.

### Rickles:

Oh, was he not a student there? He was teaching?

### Finkelstein:

Martin Davis and Jack were both students at the same time as me at City.

### Rickles:

So going back to high school, do you remember what topics were coming up in the courses, what subjects you were studying?

### Finkelstein:

The person I interacted most with somehow was the chemistry professor Liebermann. I would not be a bit surprised it’s the same Liebermann that Feynman mentions. But his was out in Long Island and mine was on Manhattan Island. But people move. They were both chemistry professors… chemistry teachers, I mean.

### Rickles:

But it was a man.

### Finkelstein:

Oh yes. Definitely.

### Rickles:

It wasn’t Hugh Liebermann was it?

### Finkelstein:

Pardon?

### Rickles:

Hugh.

### Finkelstein:

There was a Hughes at City College who was pretty important. He was my mentor in an honors course, and I believe he wrote a book with Zemansky. I think he was quite smart. He taught a course in mathematical physics at City.

### Rickles:

So when you were at high school, did you know that you were going to go to college and do physics or were you…? So what was the…?

### Finkelstein:

I was already reading… One of my folk heroes, I guess, is Gabriel Kron. Do you know the name at all?

### Rickles:

He’s the computer guy, right?

### Finkelstein:

No, no. He was an electrical engineer with mathematical and generalist inclinations.

### Rickles:

That was the guy who traveled around the world after…

### Finkelstein:

No. His specialty was applying tensor analysis to electrical networks and rotating electrical machinery. He wrote a few books on the subject which actually involved differential geometry or equations at least, more general than those of general relativity. His colleagues laughed at him for calling it tensor analysis because they were all used to matrix algebra for the purpose of networks. So he appealed to Banesh Hoffmann to verify that he was really doing tensor analysis using transformation theory, and he was.

### Rickles:

It’s not a bit like Roger Penrose did later, is it? Penrose had some kind of notation.

### Finkelstein:

Oh, it’s a very different order of magnitude. Kron had great aspirations, but his mathematics were very limited.

### Rickles:

It’s Kron, as in K-r —

### Finkelstein:

Kron, right.

### Rickles:

Yeah, I know the name.

### Finkelstein:

And Penrose is a really gifted mathematician.

### Rickles:

So when did you stumble across Kron?

### Finkelstein:

In my last year at City, I think. Yeah. Actually, it was indirect. There was a branch of Barnes & Noble near Stuyvesant High School, and wandering through it, I saw a couple of books by Doherty and Keller, the Mathematics of Modern Engineering. In it, they described Kron’s method.

### Rickles:

Yeah, because I thought Kron didn’t have a university position. I thought he was kind of independent.

### Finkelstein:

Oh no, he doesn’t. No, he didn’t. He was an engineer at General Electric in Schenectady.

### Rickles:

Ah, yeah. That’s the same guy. I know the name. Did you stumble across these books by the Liebers?

### Finkelstein:

Oh, they were delightful, yes. I enjoyed them very much. The mathematics was just at the level I could enjoy. I’m sorry they’re out of print. They’re really —

### Rickles:

Some of them are back in print, actually.

### Finkelstein:

Oh, that’s good news.

### Rickles:

Yeah, the relativity one and Infinity.

### Finkelstein:

Excellent.

### Rickles:

When did you stumble across those? Do you remember?

### Finkelstein:

In high school.

### Rickles:

In high school, yeah. Stanley Deser mentioned those as well as being sort of influential.

### Finkelstein:

Right. Books were scarcer back then.

### Rickles:

Yeah. You know she went to Robertson to proofread that book, so it has good pedigree.

### Finkelstein:

I didn’t know that.

### Rickles:

I found some files. I went through the Robertson archives. I don’t suppose you were aware of her institute? Lillian Lieber had an institute called the Galois Institute.

### Finkelstein:

I saw that mentioned in her book. It rang no bells at all. I never came across it again.

### Rickles:

Yeah. It’s very hard to find information on, actually, which is strange.

### Finkelstein:

Lovely title. Then as I approached graduation, I went to the head of the physics department there, a man named Alexander Efron, and told him I liked physics and mathematics. What should I do for a living? I gather that he graduated in physics in the middle of the Depression and was never able to find a physics job and ended up teaching high school, and he urged me not to go into physics. He said if I like physics and mathematics, I should be an electrical engineer. So I enrolled at City in electrical engineering, and really it was there that I discovered there was such a thing as a profession of physicist. Again, my first year I took my courage and switched.

### Rickles:

What did your parents think of the career choice?

### Finkelstein:

They were very permissive. They thought I was going to be a doctor. That’s why I took Latin in high school.

### Rickles:

How were you funded as an undergraduate?

### Finkelstein:

I lived at home, and my parents supported me. City College was a subway college. I lived in Harlem.

### Rickles:

So you mentioned Martin Davis and Schwartz were there again, and they were drifting into — Were they studying mathematics, both mathematics? I’m wondering if you got any of the influence of the logic and these kinds of things.

### Finkelstein:

Intense, really. They would lecture me on Gödel’s theorem until they began to understand what was going on, and Post had them programming computers before they existed.

### Rickles:

Ah. Interesting.

### Finkelstein:

To carry out his program in decidability and computable functions.

### Rickles:

What computers were they?

### Finkelstein:

All paper, conceptual. You don’t have to have a computer to program one.

### Rickles:

Excellent.

### Finkelstein:

They were doing recursive function theory.

### Rickles:

As undergraduates.

### Finkelstein:

For example, they would program a Turing machine, which is completely conceptual. I know that influenced me.

### Rickles:

So when did you switch to physics? Physics or maths?

### Finkelstein:

I switched from engineering to physics in that first year at City.

### Rickles:

To physics. You didn’t get tempted by maths?

### Finkelstein:

Tempted, but not significantly tempted. I was too much in love with physics.

### Rickles:

So what kind of discussions were you having at this time? I mean so John Stachel was there. Were you friends with John Stachel?

### Finkelstein:

Yes, inasmuch as I was friends with anyone. Stachel was certainly in my group.

### Rickles:

Do you remember anything about the… because obviously the Communist paranoia started then. His dad must have been indicted roundabout the time he started, I suppose.

### Finkelstein:

Right, and I didn’t get involved in it. But I knew it was going on.

### Rickles:

I don’t suppose you have notebooks and stuff from your undergraduate days.

### Finkelstein:

From City College? No. From graduate school maybe.

### Rickles:

Stachel mentions that the head of physics at City College wasn’t very nice. He didn’t like him. Do you remember anything along those lines?

### Finkelstein:

Let me remember.

### Rickles:

He didn’t give me a name. Because he ended up leaving, right?

### Finkelstein:

First of all, I made a mistake in names when I spoke of Hugh — when you mentioned Hugh and I switched to Hughes. The best theoretical physicist at City College at the time was Professor Wolfe. I don’t know why I called him Hughes.

### Rickles:

So who else was teaching in physics at that time in City College?

### Finkelstein:

Well, the ones I remember are Wolfe and Zemansky, and it will take me awhile to dredge up other names. I took an electronics laboratory, but I forget the name. A lovely man, but I forget his name. They even had a course in AC machinery. Electrical lab. I probably took that.

### Rickles:

So you started just as the War finished. I mean did you notice the effects of that again? Do you remember anything about the effects on the courses, on what kind of physics was done, on the teachers?

### Finkelstein:

I’m sure they were… I’m not sure they were strong, actually. They may have been. No, I didn’t notice what was going on.

### Rickles:

Do you have any recollection about what kind of physics you wanted to do whilst you were an undergraduate?

### Finkelstein:

That was really determined for me in my first year at City. I knew that one had to know quantum theory to do physics, and I found there weren’t that many books on quantum theory then. I found them incredibly opaque.

### Rickles:

Do you remember which ones you were reading?

### Finkelstein:

Well, I looked at Eyring, Walter, Kimball, and I looked at Pauling, and I found them impossible to read. The break from mechanics and electrodynamics was just too much to follow. Mathematics was not the problem. The problem was the disconnection from the mechanical way of doing physics. I really was in despair. I went into a fairly great depression. It was just an incredible stroke of luck — and also I spent some time searching — but I came across a paper by Birkhoff and von Neumann, “The Logic of Quantum Mechanics.” Actually, all the basic ideas in that paper are already in von Neumann’s book on quantum mechanics, but I didn’t know that at the time. I thought this was the beginning. Von Neumann made it very clear and convincing that quantum theory was a change in logic corresponding to that of the change in geometry and general relativity. Within a day or two I realized that if one changes logic, then one must reconstruct the whole set theory which was founded on that logic. I really had some electricity in my spine at being in on the bottom of that ladder.

### Rickles:

So do you think your interactions with Davis and Schwartz sort of helped prepare you for taking that on board?

### Finkelstein:

I’m certain of it, really.

### Rickles:

Did you discuss the von Neumann paper with those or with anybody else?

### Finkelstein:

I probably tried to. Of course it was what I was thinking of all the time. But I never found anyone who was responsive.

### Rickles:

Yeah. I’m trying to remember what the aftermath of that paper was. I don’t think it got heavily cited, really, did it?

### Finkelstein:

I think it is now.

### Rickles:

It is now, yeah, because of the idea that logic is an empirical thing. But yeah, at the time.

### Finkelstein:

Actually, there’s nothing really new conceptually beyond what Heisenberg said. What von Neumann was doing was axiomatizing Heisenberg’s statements. Heisenberg already understood that filtrations don’t commute in quantum theory and they do in classical physics. It’s just that von Neumann had rather recently finished his thesis in which he invented a new functional set theory, so he was alert to axiomatizations.

### Rickles:

Were you not tempted to go and try and study with either of these, von Neumann or Birkhoff?

### Finkelstein:

I’ll say that never occurred to me.

### Rickles:

Did you ever meet von Neumann later on?

### Finkelstein:

I chauffeured him. When I was a graduate student at MIT, he came to town to give an address to the American Mathematical Society of Harvard, and I was the graduate student they entrusted with getting him from Tech Square to Kendall Square. I didn’t have the courage to say a word.

### Rickles:

[Laughter] That’s very good.

### Finkelstein:

But by the way, his talk was one… He was asked to update Hilbert’s problems, as any problems of mathematics today, and he really only gave one: exploring quantum mathematics. The content was his quantum logic.

### Rickles:

I take it you went and saw the lecture.

### Finkelstein:

I was at that lecture, yes.

### Rickles:

Did you ask a question?

### Finkelstein:

No.

### Rickles:

Okay, so yeah. So why did you —

### Finkelstein:

And looking back from my present understanding, I was incredibly naïve in those days, and I realize that von Neumann was too in that he dropped the operational foundations that Heisenberg had and really concentrated on axiomatizing one aspect of Heisenberg’s work. So it’s influenced more by his own background, and I would say a belief in matter waves, than by the philosophy of Heisenberg and Bohr. But he was quite clear that the old logic was gone and the new one worked.

### Rickles:

You mean the focus on observables.

### Finkelstein:

He didn’t talk about observables. I don’t think there’s a description of an experiment.

### Rickles:

No. That’s true, actually.

### Finkelstein:

It gave an opportunity to use Hilbert space. There’s this famous paper by Wigner on “The Unreasonable Effectiveness of Mathematics,” and then there’s this response by Jack Schwartz, “The Pernicious Influence of Mathematics in the Natural Sciences.” This book by von Neumann is an example of both.

### Rickles:

Yeah, I know those papers well. So did you study — Who was teaching quantum mechanics at City College?

### Finkelstein:

I took an honors course with Wolfe. Well, actually I took one with Wolfe and one with Zemansky.

### Rickles:

How do you spell Zemansky?

### Finkelstein:

Z-e-m-a-n-s-k-y. With Zemansky, I read this very, very fat book on quantum statistical mechanics. I’m blocking the name because I haven’t looked at it in 50 years or longer. Tolman. Tolman’s book.

### Rickles:

Oh yeah. I assume you didn’t do any general relativity. Was there any general relativity being taught at all?

### Finkelstein:

No, but I read Einstein’s book. In my early years at Stuyvesant, at Stevens, thanks to the generosity of Peter Bergmann, I went to a conference in France at L’Abbaye De Royaumont, not far from Paris. That was my first conference in general relativity.

### Rickles:

When was that? That must have been ‘58, ‘59 though, right?

### Finkelstein:

Right. Quarterly, and I’m trying to remember what I was doing there.

### Rickles:

Did you give a talk? Did you give a paper?

### Finkelstein:

I think so, yes. It was ‘59. No, I don’t think I had done the work I did on the black hole by the time of that meeting.

### Rickles:

There must have been an earlier meeting, I think, because I think you gave —

### Finkelstein:

I’m not sure what I talked about. Anyway, how did we get on to that? Oh yes. This was my first introduction to a group of relativists. I went around asking people my age why they took up relativity, and without exception, all of them gave the same reason I did: Eddington’s book The Mathematical Principle of General Relativity.

### Rickles:

I was going to mention Eddington. There are a few — When do you read Eddington’s book?

### Finkelstein:

When I was at City College.

### Rickles:

Did you have a go at any of other Eddington’s —

### Finkelstein:

Oh, I read it from cover to cover, yeah.

### Rickles:

Did you read any of his other books?

### Finkelstein:

I went to his other books also. I tried to read __Fundamental Theory__.

### Rickles:

Yeah. Deser says the same thing. He tried to read __Fundamental Theory__. [Laughs] Did you try again later on in life to read that book?

### Finkelstein:

Well today, I have a little more… It’s not just that I don’t understand it. I see him making important mistakes. But they’re deep mistakes. It’s not mathematical errors.

### Rickles:

Did you learn from Eddington’s book with anybody or was this just on your own?

### Finkelstein:

Yes.

### Rickles:

When? At City College or MIT or later?

### Finkelstein:

Oh, it certainly wasn’t MIT. It must have been City, yes.

### Rickles:

Okay. Did you choose any mathematics courses to help you learn this kind of material?

### Finkelstein:

The mathematics course…First of all, Feshbach gave a course in mathematical physics using a mimeograph version of what later became his book, Morse and Feshbach.

### Rickles:

That was at MIT, though, right, not at City College.

### Finkelstein:

Yes. At City College, I can’t remember taking any important math courses that weren’t required.

### Rickles:

Rarita wasn’t at City College, was he?

### Finkelstein:

Didn’t meet him.

### Rickles:

Was he there, though?

### Finkelstein:

He was ahead of me, I believe. Yeah. At MIT, I took a couple of math courses that were important for me. One was under Walter Harry Pitts.

### Rickles:

Yeah, I was going to mention him.

### Finkelstein:

He taught a course out of Stone, functional analysis. It was basically a course in Hilbert’s space algebra.

### Rickles:

That’s not Pitts as in McCulloch and Pitts?

### Finkelstein:

It is. Exactly.

### Rickles:

It’s that one.

### Finkelstein:

We were very good friends, to the extent that either of us were able to have friends at MIT. We went for walks along the beach. I remember him picking up clams from the sand and eating them. This struck me as odd at the time, but of course the same clams you get in the fish market, so why not? He introduced me to McCulloch.

### Rickles:

So they’d done their… because I didn’t think he —

### Finkelstein:

I spoke with him of my interest in general relativity, and he said, “But that’s merely special relativity made local.”

### Rickles:

What was Pitts’s background? Because I always think of him as the McCulloch and Pitts person and the nervous system stuff. But he was a mathematician or a mathematical physicist.

### Finkelstein:

He was a protégé of Wiener, and he was I think a real genius, self-taught. His father worked in a boiler factory making boilers, and I think Pitts did too for a while. It’s a tragic story. He learned enough chemistry to make his own drugs and destroyed his brain. Quite a few years later when I was at Stevens, I made a pilgrimage. I realized how important my acquaintance with him had been, and I made a pilgrimage to Cambridge to say thank you. I found him finally, and he was sitting reading in the bar in Harvard Square. He didn’t remember me, and he wasn’t really able to carry on a conversation and wasn’t interested in carrying on a conversation. So the Pitts I knew was gone. I don’t know how much longer he lived after that.

### Rickles:

I read a book about Norbert Wiener, a biography, Norbert Wiener. Apparently there was some issue between him and McCulloch, I think, which led onto Pitts getting depressed. Maybe I’m remembering it wrong. Did you remember anything about that interaction? Where you there when Wiener and McCulloch and Pitts were all there at MIT?

### Finkelstein:

Yes, certainly. Absolutely, yeah. I wasn’t involved in their direction.

### Rickles:

So going back to City College, did you do a bachelor’s thesis while you were there? What was that on?

### Finkelstein:

Right. Our teacher was a delightful man. Jack and I took an honors course together over the summer, and we had to produce a thesis. His name will come to me in a moment. He was a little odd in that he had an infinite number of children. He furnished his room with stacks of cots.

### Rickles:

Yeah, that would make you a bit odd.

### Finkelstein:

They ran him ragged. There was nothing like parental discipline. His name is on the tip of my tongue. I haven’t uttered it in decades.

### Rickles:

Do you remember what he taught?

### Finkelstein:

Jack’s honors thesis was on vector spaces with a transfinite number of dimensions beyond Hilbert’s space, which has a countable number of dimensions. We had read Andre Weil’s book together during that summer in our intensive way, and I wrote a thesis on alternatives to Weil’s concept of algebraic geometry.

### Rickles:

Weil… when you say Weil…

### Finkelstein:

Andre Weil.

### Rickles:

Oh, as in W-e-i-l, yes.

### Finkelstein:

Right. It was in that study that I learned the importance of algebra as being regular.

### Rickles:

So you’d learned some algebraic geometry tools while still an undergraduate.

### Finkelstein:

I guess so. Yes, right. That turned out to be very useful in reading quantum logic because of the use of algebra in setting up coordinates for quantum logics.

### Rickles:

Did any of that go into your thesis or was it separate?

### Finkelstein:

There was no mention of quantum logic in my thesis. It was pure algebraic geometry.

### Rickles:

But did you have it in mind? No?

### Finkelstein:

I didn’t realize how useful it would be.

### Rickles:

Okay. So when did you decide on MIT? Did you apply various places or was MIT the only place?

### Finkelstein:

I probably applied to a few places. I know I applied to Princeton, but they had a quota.

### Rickles:

Did you have anybody in mind at Princeton or just…?

### Finkelstein:

Not really. I knew what I wanted to do. I guess you would call it quantum gravity. But they needed someone to do computation in the astronomy department, and they hired one of my best friends, also going back to Stuyvesant High School and then City, Irvin Rabinowitz. That filled their quota for New York Jews for that year. So I went to MIT, and I’m very grateful I did.

At MIT, I was allowed into Weisskopf’s evenings with his graduate students, an experience that I’ll always cherish. Marvelous man. He had an attitude to quantum theory related to that at Princeton like the difference between Heisenberg and Wigner. So I learned it almost from the horse’s mouth, and I’m very grateful. At Princeton they were worried about things like collapsing wave functions and multiple universes. But they were certainly going around saying that nobody understands quantum theory, and the fact is Heisenberg understood quantum theory and Bohr came very close. The big problem was getting out from under the idea of matter waves, the ancestry of de Broglie and Schrödinger, who never understood quantum theory. He wrote papers to prove it like his famous Schrödinger cat paradox. Nowadays computers have Schrödinger states, and they are as close to absolute zero as they can get them. So for a cat to have a wave function, you have to have a beam of cryogenic cats, so-called that you can do interference experiments with it. Then the question of whether the cat is alive is kind of silly.

### Rickles:

Yeah. I think Anton Zeilinger once said he wants to put himself through a quantum interference device one day!

### Finkelstein:

Right.

### Rickles:

Did you begin with a master’s degree at MIT or was it straight into —

### Finkelstein:

At MIT, the master’s was given en passant. If you decide to stay in for doctorate, they gave you a master’s anyway.

### Rickles:

Yeah. Weisskopf was your supervisor?

### Finkelstein:

No. Rickles: Or was it Feshbach?

### Finkelstein:

Or he was my supervisor until Villars came. I got there a year before Felix Villars.

### Rickles:

How do you spell that?

### Finkelstein:

Villars.

### Rickles:

Oh yes. As in Pauli and Villars.

### Finkelstein:

He came from working with Pauli, so there was a little family incest there, Pauli, Weisskopf, Villars.

### Rickles:

Feshbach was there as well, though, right? Was he there? Feshbach.

### Finkelstein:

Did you say brother?

### Rickles:

No, he was there at the same time, and he was doing similar—

### Finkelstein:

Yes, absolutely. In fact, I did a problem under him. I never took a course in quantum theory, and I got there when no course was being offered because I came in the middle of the term. I had graduated a little early from high school… from City College. So Weisskopf just gave me the book on quantum theory of… It was the standard book at the time on… Schiff. And told me to do every problem. So I handed in the problems, and that was my education in quantum theory. I’ve never taken a course in quantum theory.

### Rickles:

Is it Leonard Schiff?

### Finkelstein:

Leonard Schiff it is.

### Rickles:

That was the standard MIT textbook, was it, that everybody did?

### Finkelstein:

I guess. Right. And it’s one of the best, I still think.

### Rickles:

Did you do any quantum field theory, or was that from the Schiff book? Does Schiff go into quantum field theory? I forget.

### Finkelstein:

A little bit at the end, yes.

### Rickles:

So was your thesis topic set by your supervisor?

### Finkelstein:

Yes.

### Rickles:

Were you also working on separate projects as well at the same time or was it just —

### Finkelstein:

Well, I tried to educate myself. I must say that the period when I was doing my thesis is one of the more unpleasant times in my life. I didn’t want to work on nuclear forces. But the nonlinear theory of nuclear forces was as close as they came to the nonlinear theory of gravity, quantum in both cases. So that was the best they could do for me. But I slept 15 hours a day or something like that.

### Rickles:

So when you were given this project, was there some conscious decision to have it connected to general relativity?

### Finkelstein:

Oh yes. That was as close as they could come.

### Rickles:

So why at this stage did you want to do general relativity? Where did that come from?

### Finkelstein:

Well, that’s part of the original epiphany, namely if changes in geometry show themselves up as fundamental forces, then chances are changes in logic might take place from place to place and so as the other forces. So I had the idea of paralleling Einstein’s work in a quantum logical syntax. That actually is consistent with the gauge program. I didn’t quite understand that at the time, but I felt that what I was doing was building on general relativity with trying to move the structure from its classical foundations to quantum ones without destroying it.

### Rickles:

So you were trying to solve the problem of quantum gravity, really.

### Finkelstein:

Well, I wasn’t going to put it that way.

### Rickles:

Was it you —

### Finkelstein:

I wasn’t interested in quantizing gravity. I was interested in quantizing the universe. Gravity was just one part of it.

### Rickles:

Oh, so you’re trying to get just some kind of unified theory based on logics.

### Finkelstein:

I think I was looking for a unified theory. To me that is no longer a right thing to do. It’s a trap. But I was pretty deep in the trap.

### Rickles:

So you didn’t have any Gödel kind of problems with unified theories of everything at the time [unintelligible].

### Finkelstein:

I think it’s an archaic notion. I trace it back to the logos of Parmenides. But I was buried. I was floating in it.

### Rickles:

So who were you talking to about that side of your work while you were at MIT? Because I mean there were plenty of people who were sort of interested in similar things. So Norbert Wiener — I’ve just been at MIT, and I was looking at the Wiener archives.

### Finkelstein:

Wiener was a problem. What was his attitude? He had his own quantum theory. He deeply resented Heisenberg’s success in solving a problem that Wiener couldn’t handle. What was it again? Anyway, he would bother Weisskopf with his ideas for hours, and finally in desperation, they decided to throw him a graduate student. They offered me the position, but going back to classical physics, which is what Wiener was trying to do, was the last thing I wanted to do. They offered it then to another graduate student who collaborated with Wiener.

### Rickles:

Was that the differential space guy in quantum [overlapping voices]? I forget the guy’s name.

### Finkelstein:

It could be, yes. In a moment I’ll probably get the name of the student. But it never came close to quantum theory. It was working on… Differential space sounds right.

### Rickles:

I forget the student. I know which one you mean. I can find out.

### Finkelstein:

They published joint paper after joint paper on this idea and never got anywhere.

### Rickles:

Yeah, I can found that out. But I actually found a paper from 1951 in the Norbert Wiener archives where he was trying to come up with a unified theory of physics based on Eddington-type ideas, on the centrality of the observer. Do you not remember?

### Finkelstein:

But Eddington at least had non-commutative variables in his work. I mean he got the essence of quantum theory, and Wiener never was willing to take that path.

### Rickles:

I’m trying to think who else was there who would have been relevant. So who else were you speaking to, if not Wiener?

### Finkelstein:

My officemate was Kerson Huang. We became pretty good friends. I’m sorry that life has separated us.

### Rickles:

Was Gabor at MIT?

### Finkelstein:

Gabor. I never met him as far as I know.

### Rickles:

Because he was working on this…

### Finkelstein:

Holography?

### Rickles:

Right, and the logon stuff. There are vague similarities to trying to get some kind of quantum of information idea. You don’t remember any talks? Were there seminars — general, regular physics seminars?

### Finkelstein:

Right. There were a couple. There was a little seminar where graduate students were given the opportunity to exercise their ability before faculty. I gave a talk there on the nonlinear meson on the Tomonaga theory of bosonic excitations in a fermionic gas. I realize now that my recent work is actually relevant to the problem, how do you make bosons out of fermions? You could think of Tomonaga’s work as a limiting case.

### Rickles:

So your thesis was on the nonlinear meson theory.

### Finkelstein:

Yeah. Essentially a theory purported by Teller first, I think.

### Rickles:

So it’s not Yukawa’s theory that you were looking at or anything.

### Finkelstein:

It presupposes Yukawa’s theory. But Yukawa proposed a linear meson theory. The linearity was the interaction between the meson and the nucleon, and the free meson was linear. Teller added what amounts to a repulsive force of meson to meson to account for the saturation of nuclear forces.

### Rickles:

So had the cosmic ray stuff happened by then?

### Finkelstein:

Oh yes.

### Rickles:

So is this why it was being pursued so strongly?

### Finkelstein:

I didn’t know about the connection.

### Rickles:

Okay. So you were at MIT until 1953.

### Finkelstein:

Right.

### Rickles:

So obviously we get the Korean War during that. Again, was there any impact of the Korean War on you during that time?

### Finkelstein:

No. By then I was married, and my wife was pregnant. I was at Stevens, and we both decided that I shouldn’t leave her. So I took advantage of draft deferment that was available.

### Rickles:

Did you do any war work? Were you sort of expected to work on —?

### Finkelstein:

Yes. So for a while I actually transferred from Stevens to NYU and worked at the AEC computing facility. I was studying radiation effects in the upper atmosphere.

### Rickles:

I have that you did some work, and I think the photo I’ve got is from when you were working on —

### Finkelstein:

That’s lab work at Yeshiva University. That comes a bit later in the story. What happened is as a result of my work at what later became the Courant Institute at the AEC computing facility, yes, I was even hired to work for Project Sherwood. That’s the name associated with the Los Alamos group, but I guess somehow the NYU group fitted under that rubric. Well, at Princeton…

### Rickles:

Matterhorn.

### Finkelstein:

…it went under another name.

### Rickles:

Do mean the other project?

### Finkelstein:

I’m trying to remember the name of the Princeton Plasma Lab.

### Rickles:

Yeah, it was Matterhorn.

### Finkelstein:

Matterhorn, yes.

### Rickles:

Lyman Spitzer.

### Finkelstein:

Exactly. Spitzer, Matterhorn, Jim Tuck of Sherwood Forest.

### Rickles:

I’ll maybe come back to that. But you —

### Finkelstein:

I probably said Schpitzer [mispronounced]. I meant to say Spitzer. He’s an American.

### Rickles:

Yeah. Because I have that you were doing some computing work in 1953 at MIT on Project Whirlwind. Is that right?

### Finkelstein:

Yeah. That was my thesis. Actually, it was two steps. First, I had a nonlinear meson theory. I had to solve the radial equation, so I needed a computer to put it on. First I put it on Vannevar Bush’s differential analyzer, and it kept going off scale before I got to the origin or anywhere near it. I thought this might be a computational fault, so I switched to Whirlwind and ran the computer there, ran the program there, and again, never got close. I could not approach the origin. I expected to find some function like that you call a potential or the Coulomb potential all in one defined on the whole real axis. But I couldn’t get below a certain point. Finally, a good high school friend, Donnie Newman, sort of took me aside and told me the facts of life about nonlinear differential equations and wandering singularities. So the radial equation for the Schiff nonlinear meson which has a Φ^{4} term after the Φ^{2} term with a + sign, the real equation, that blows up before you get to the origin at a point which depends on the amplitude of infinity. That was my first encounter with the wandering singularity, and I met it again in plasma physics, and I met it again in the black hole. Every time it came as a surprise.

### Rickles:

How were you able to get on Project… on the computer? Was it just available? Did you have to sign a thing saying, “I would like to use the computer for this long”?

### Finkelstein:

I think there was no difficulty. I simply walked in and asked, “Could I run this?”

### Rickles:

How did you know how to write the programs for it in those days? Who taught you that?

### Finkelstein:

Oh, I don’t remember. They probably just gave me the manual. I’d already been programming on Turing machines and so on, thanks to Post.

### Rickles:

Imaginary Turing machines!

### Finkelstein:

Right. Oh also, while I was in high school, I tried to build a logic digital computer in a garage. We had no — It was me and Virgil Johannes, who later became the head of an electrical engineering department at a school in New Jersey. It involved rectifiers. I tried making them out of copper oxide — out of copper dipped in acid — and I couldn’t get good enough coatings. It never got off the ground. But anyway, it relieved a childhood itch to actually be near a computer that worked.

### Rickles:

That’s very good. [Phone rings.]

### Finkelstein:

It will go away. They always stop.

### Rickles:

[Laughs] Do you think getting your hands on computing equipment had any sort of influence on the type of logical work that you did and the nature of —? You know Bryce DeWitt works on the computers. He was on Project Sherwood, wasn’t he? He had the Lagrangian hydrodynamics stuff, detonation hydrodynamics.

### Finkelstien

Right.

### Rickles:

You can sort of trace lines from that work into his quantum gravity work. Do you think there’s anything traceable from the computing side in your case?

### Finkelstein:

I don’t know. I’m not conscious of a strong connection. But I’ve always felt at home with computers because it was early study with friends who worked under Post.

### Rickles:

So it looks like you wrote a software package with somebody called Gill and Rotenberg called SADSAC. Do you remember that? How did that come about?

### Finkelstein:

It just happened that after I finished my thesis work, there was a crash program to produce a package that could be used by executives of industry and finance who wanted to explore the use of computers in industry and business. Whirlwind had a totally unfriendly language, and FORTRAN didn’t exist yet. So we put together a single-address simulated automatic computer that was much friendlier than the assembly language, machine language. It had instructions that any business user could use and gave an automatic post mortem whenever there was a crash.

### Rickles:

So it debugged itself basically. Was that the first one to do that? Was it the first example of that?

### Finkelstein:

I would not be surprised. I’ve never checked. Stanley Gill was over from England. He had worked with Wilkes.

### Rickles:

Oh yeah, Maurice Wilkes. So who came up with the name SADSAC?

### Finkelstein:

I probably did. I like names. The thing is that while Whirlwind ran like the wind, we so slowed it down with our post mortem facilities.

### Rickles:

The early version of Windows.

### Finkelstein:

It was run interpretively, so you could actually hear the computer doing instructions one at a time — thump, thump, thump.

### Rickles:

Did you meet Jay Forrester or anything? So he wasn’t in charge or anything?

### Finkelstein:

But I saw a room full of Forrester tubes not long before they disappeared and were replaced by one little box of…

### Rickles:

Yeah, those little magnetic loops.

### Finkelstein:

…of loops, right. That was impressive.

### Rickles:

I mean did you try and search anything out about people working on applying computers to physics? Because Konrad Zuse, when was he doing his…? Oh no, that was much later, I suppose, Konrad Zuse. Okay, so ‘53. Then you went to Stevens in ‘53, or did you go to Stevens earlier and then there was some back and forth?

### Finkelstein:

The only back and forth was with NYU.

### Rickles:

Right. So remind me what the NYU —

### Finkelstein:

I had moved from Stevens to NYU, and then I moved back.

### Rickles:

In ‘53 or later?

### Finkelstein:

I probably spent about a year at NYU. In that time, I visited the Livermore Radiation Lab and worked there with Windy, Winston Harry Bostick.

### Rickles:

Okay, so Bostick was head of Stevens, wasn’t he, later?

### Finkelstein:

Not was. First I met him at Livermore, and then when I returned to Stevens, I suggested him as head of the department, and he accepted.

### Rickles:

Okay. Was Bryce DeWitt at Livermore at the time?

### Finkelstein:

I have no idea. I didn’t realize he was associated with Livermore. I don’t think I met him there. I would remember that.

### Rickles:

Yeah. Well, maybe he’d left by then. So who else was at Livermore? How long did you go to [overlapping voices]?

### Finkelstein:

It was a summer.

### Rickles:

Was it just Bostick who you hung around with or were there—

### Finkelstein:

I interacted with Bostick more than anyone.

### Rickles:

Did you start to discuss general relativity things with Bostick? Because he had an interest, right?

### Finkelstein:

He was an interesting man, but he was not a mathematician at all. Even his plasma physics was pretty much intuitive. He visualized magnetic lines of force. He did end up with a unified theory. He thought that everything was a magneto plasma.

### Rickles:

Yeah, because he applied… There’s a paper of his in the Gravity Research Foundation essay competition thing from 1958 where he had a theory of the stabilization of elementary particles by gravitational forces, which I assume sounds plasma-ish. Yeah, so that’s where that comes from. So who did you suggest Bostick as head to?

### Finkelstein:

I guess to the administration.

### Rickles:

Who was head before that? Did it have a head or was it running with a —

### Finkelstein:

What happened to the previous head? I don’t remember.

### Rickles:

So how did you get hired at Stevens?

### Finkelstein:

How did I get hired? Through a friend. George Yevick had gotten his degree at MIT and gone to Stevens. I always knew him quite well and recommended me to Yevick. We became good friends. He’s deceased now.

### Rickles:

Did you receive any other invitations from other places for jobs or anything?

### Finkelstein:

No. No, I guess perhaps because my work wasn’t that great or I’m isolated or something. But I had to pretty much make my own way. The thing is I hadn’t worked on the real problem of any member of the faculty, and so they couldn’t recommend me to their colleagues. There wasn’t a quantum gravity community.

### Rickles:

So what was your —

### Finkelstein:

But I consider myself very fortunate to get a teaching job which allowed me enough time to think.

### Rickles:

Well, and it was lucky that it was at Stevens obviously because it became the relativity hub.

### Finkelstein:

Well, that came after my going there, of course. I invited Schiller and…

### Rickles:

Anderson?

### Finkelstein:

I don’t think I had anything to do with Anderson’s coming. I don’t remember his connection. But maybe he came because of Schiller. Schiller was already doing relativity because of Bergmann. He had his credentials, and Anderson, I don’t know who he worked for, but he was already a relativist.

### Rickles:

Well, yeah. He was at Syracuse as well.

### Finkelstein:

Was he?

### Rickles:

Yes, so he was under Bergmann as well.

### Finkelstein:

Excellent. So that became a little offshoot of Bergmann’s of Syracuse, and then they are responsible for the meetings. I was just fortunate enough to be around.

### Rickles:

Let’s see. So what was your initial research program? Do you remember? When you started at Stevens, what was the thing you were interested in at the time?

### Finkelstein:

I felt I had to do an apprenticeship before I tackled the real problem. I certainly went in through classical field theory, but Einstein talked as if a theory like general relativity, like gravity, might have all the elementary particles in it if you look carefully. To the quantum mechanic, this raises immediately the question of spin-1/2. How do you get spin-1/2 out of gravitons which have spin-2? But here my reading of Wigner was crucial. Wigner gave a topological theory of which systems could have spin-1/2. Not really. He explained the spin-1/2 of the electron as a result of the double connectedness of the rotation group. Because of that, I took a course under Hurewicz at MIT on topology and learned about the higher homotopy groups from the man who invented them.

### Rickles:

Who?

### Finkelstein:

Witold Hurewicz.

### Rickles:

Oh yeah. Hurewitz I always pronounced it.

### Finkelstein:

Right. Hurewicz. He was a very dear man. I realize that in order for the gravitation field to have spin-1/2, you have to figure out how the rotation group acts on it and then see if the orbits of the rotation group can be shrunk to a point or not. There are only two kinds: those that you can and those that you can’t. If they can, then it doesn’t have spin-1/2, and if they can’t, then in principle it could have spin-1/2, even though that might get lost in the perturbation theory calculation. Fundamentally the theory is nonlinear, so the topology could be non-trivial. You have to do the calculation of the homotopy group. I think this was probably the first application of higher homotopy to physics. Nowadays they speak of moduli spaces.

### Rickles:

When are we talking here? Is this when you first started at Stevens or earlier when you were doing…? So ‘53?

### Finkelstein:

I started at Stevens in ‘53.

### Rickles:

And you were immediately working on that idea, getting spin-½?

### Finkelstein:

Right, and fortunately, I met Misner at these relativity meetings. He was at Princeton which was a hotbed of homotopy theory, and he had studied it there. He used homology theory intensively in his work on electromagnetism in gravitational fields in the formulation of the theory and in general relativity, too. But he knew homotopy theory as if he had worked in it. So I went to him with my problem. This was one of the thrilling moments of my life. He just turned around and pulled Steenrod’s theory of fiber bundles off the shelf and in a few moments converted my question about the homotopy of the gravitational field to a question about the homotopy of manifolds of subspaces. Nowadays there’s a name. No, there already was a name in Steenrod. I’m blocking on it. Anyway, the problem was reduced from finding the first homotopy group, the fundamental group of the field, to finding the fourth homotopy group of finite dimensional manifold mainly of light cones in Minkowski space-time. This is based on the fact that the topological purposes, the gravitational field was just a field of light cones. The answer came out two. You had to go around twice. There were two elements to the homotopy group. It was non-trivial. That meant that the gravitational field could have spin-1/2, even if Einstein thought so. For a moment I was tempted to start studying the gravitational field, but of course that assumes classical space-time, and I knew that one had to get to quantum space-time to carry out the program. So I resisted the temptation. But that led to the black hole.

### Rickles:

Oh, we’ll come to that in a second. How did you know about Misner?

### Finkelstein:

We met at the Stevens meetings.

### Rickles:

So when did they start, the Stevens meetings?

### Finkelstein:

Shortly after I arrived.

### Rickles:

Really? They’ve been going that long? Was this the general seminars?

### Finkelstein:

Yes. No, no.

### Rickles:

Because then there were —

### Finkelstein:

No, not general. It wasn’t a Stevens seminar. It was organized by Anderson and Schiller.

### Rickles:

Oh. So they go way back.

### Finkelstein:

I’d probably been there a couple of years at the time.

### Rickles:

I have the first one starting in 1958, but you think they were way before that. I mean there was —

### Finkelstein:

It’s possible. From this distance, time telescopes. There weren’t — It’s true I didn’t go to that many. You’re probably right. That’s consistent with when I published on the black hole and so on. It was all within a couple of years of that.

### Rickles:

So let me see if I can find it. I have you giving a talk at the Stevens meetings on…

### Finkelstein:

It’s probably just I’d been thinking about these things for longer. But I certainly met Misner at those meetings.

### Rickles:

So in November 1957, there’s you giving “Spin without Spin.” Maybe that’s something different. “An Ordinary Unified Theory Simplified by Spinors.”

### Finkelstein:

Oh yeah. I think I did… yes, as rehearsal for the spin of gravity, I looked at other classical models for spin-1/2. I asked what kinds of bodies could by their rotation have spin-1/2. That was another warm-up exercise.

### Rickles:

Yeah, because your first paper in ‘55 was on the internal structure of spinning particles.

### Finkelstein:

That’s right. I asked what kind of particles acted on by the Lorentz group — particles not in the modern sense, but in the sense of structures. What invariant objects could have spin-1/2? For example, the sphere can’t because it doesn’t have enough markers. Rotation of the sphere is trivial. A tetrahedron with relativistic analog or pentahedron could if you mark vertices so you can tell they’ve turned around. I came up with a handful, or a small handful, of theories that can have spin-1/2. For example, a field described by a null electromagnetic field, *E+iB*, the field of a light wave, if you just imagine an object with that attached to it, it could have spin-1/2 if they’re orthogonal, if it’s a null vector.

### Rickles:

So where do you think that —

### Finkelstein:

And of course if you apply tensor analysis, the usual, if you just look at the indices, you wouldn’t think it could have spin-1/2. So that was encouraging.

### Rickles:

Actually, I have that you did a paper called “On Relations between Commutators.”

### Finkelstein:

That’s different. That grew out of my interest in quantum logic.

### Rickles:

Right. So the past —

### Finkelstein:

I thought everything had to be done non-commutative, and the first thing you learn is calculus. So you’re getting non-commutative calculus, and the core of that was provided by H. S. Green, who at that time I believe was in Australia already. I pulled that out of his paper and expanded it into a proof of the well-known theorem about expanding a product of two exponentials as an exponential of the name of that theorem that I’m forgetting. It comes up again and again in the old parts of quantum field theory. E^{a}xE^{b} = E^{a+b} + a half the commutator of a and b and so on into higher powers. Of course the name is in the paper.

### Rickles:

I’ll check that. Okay, so that was kind of a separate research project then, whereas the —

### Finkelstein:

It’s all along the line of trying to quantize everything that moves, including the kitchen sink.

### Rickles:

So the “Past-Future Asymmetry of the Gravitational Field” paper, so that was ‘58.

### Finkelstein:

That came out of the spin-1/2 gravity [overlapping voices].

### Rickles:

Was that in the discussion with Misner, or was that a different paper than Misner?

### Finkelstein:

I discovered the spin-1/2 with Misner definitely, and then I looked at the simplest model, the simplest gravitational field whose rotation could do that, and it looked like this. Here are the light cones at infinity, and they tilt toward the horizon like this, and then they meet there. If you have this happening in every radius so it’s spherically symmetric, then that generates the homotopy group of the gravitational fields. Every gravitational field is made of these and their antiparticles, the time reversals. Looking at it, it was clearly not symmetric under time reversal. You could see that because you could go in, but you couldn’t get out. That really scared me because this Birkhoff theorem which says that if it’s stationary and spherical at infinity, it’s static. No difference between past and future. But that’s a folk theorem. If you look carefully at Birkhoff’s theorem, one way to say it is that if it’s spherical and static at infinity, spherical and stationary to infinity, it may static down to the first point where the time axis is a generator of the light cone, down to the horizon. So Birkhoff’s theorem already says that if you look for coordinates that makes it static everywhere, it will blow up at the horizon. If you want to go through the horizon, you just have to go back to a coordinate system where the light cones are allowed to rotate. So he put in — That means a cross term in space and time. So he put in a variable one of the simplest kind and just adjusted it to eliminate the singularity of Schwarzschild. It took a few minutes. So that’s a byproduct of the search for spin-1/2 in gravity.

### Rickles:

So it mentions you discuss that with Stachel and Anderson and Bruce Crabtree and Melvin Hausner as well. So were they all at Stevens at the time?

### Finkelstein:

Hausner was at NYU. Stachel was my graduate student at Stevens. Who else did you mention?

### Rickles:

Bruce Crabtree.

### Finkelstein:

Bruce Crabtree was on the faculty at Stevens.

### Rickles:

Yeah, and Anderson as well.

### Finkelstein:

And Anderson, too.

### Rickles:

I should have asked him. What was Stachel working on? He was doing a master’s thesis, wasn’t he?

### Finkelstein:

Right.

### Rickles:

Was it a general relativity problem?

### Finkelstein:

I’m trying to remember whether I inflicted quantum logic on him now or…

### Rickles:

He has a paper on quantum logic somewhere.

### Finkelstein:

I don’t think he liked — no, he didn’t like the idea of quantum logic. I’m sure he didn’t write on it. For one thing, real Marxists think that quantum theory is idealistic and conflicts with the doctrines of Marx. Namely, the goal of physics is to control the universe, and if Heisenberg were right, there would be limits to that control. If you meet limits, you simply have to try harder. I don’t know if they had the same problem with relativity and the speed of light. I think Stachel has completely healed since then.

### Rickles:

[Laughs] He seems to. He seems to have. I’ll try and get the chronology right. So in fifty… So Misner was working on this Rainich theory. Did you discuss the Rainich theory? He already [overlapping voices].

### Finkelstein:

No, we didn’t. I knew of his work on it.

### Rickles:

So I’m just trying to work out when you must have met Misner because I think it must have been…

### Finkelstein:

It was certainly an important meeting for it. Let me think a minute.

### Rickles:

It must have been earlier because I remember Misner and Wheeler had this “Annals of Physics” paper on geometrodynamics in ‘55, but it would have been written in ‘54. That had all the homology stuff in it, and he was doing Betti numbers and all this kind of stuff. So maybe would you have read that, do you think, when it came out?

### Finkelstein:

I’m sure I read it, yeah.

### Rickles:

And then I’m just wondering if there were already some sort of back and forth between Princeton and Stevens because Wheeler gets interested in relativity in about ‘53, right? Quite late.

### Finkelstein:

I’m sorry?

### Rickles:

John Wheeler started looking at relativity in about 1953.

### Finkelstein:

Right.

### Rickles:

He was sort of traveling all over the place trying to find things, so I’m just wondering if he went over to Stevens quite early on and maybe took Misner with him or something.

### Finkelstein:

I don’t remember him coming to Stevens except for the relativity meeting, but it could have escaped my memory.

### Rickles:

Okay. So do you remember also Misner’s Feynman quantization approach? Did you have any interest in —

### Finkelstein:

Oh yeah, I did of course. I studied that very carefully. I read both Dirac’s and Feynman’s papers.

### Rickles:

So the way you look at that work, is the idea to get to use the multiply-connected space-time to get the right kind of particles there? Because I’ve often wondered why he was doing Feynman quantization.

### Finkelstein:

Of course when Feynman wrote, he built on classical space-time, and that meant that a lot of his mathematics were meaningless. You can’t really do those intervals over the paths. I think of that more as a research program than a theory, and in fact, he discretized his space and time in order to get answers, and then went to the limit of the continuum of zero spacing. I’ve wanted to… I think that space-time is built of atoms — quantum atoms, of course. I even named them in anticipation. So all the procedures of renormalization I consider to be kind of irrelevant. Renormalization will survive as a physical process just as you have the [?] shielding in a plasma. That’s a renormalization of charge. But it won’t be infinite.

### Rickles:

So on the path — Since you had this… Did you already have the discrete idea at Stevens then?

### Finkelstein:

Oh, I don’t think it was discrete ever. I think it was quantum, which is neither discrete nor continuous. I think in terms of a Hasse diagram or a lattice diagram, and in classical physics, they’re exactly as high as they are wide. If you have a coin flip, it’s… the diagram is a square empty — the false, the true, and the two heads and tails. So it’s two wide and two high, and that’s true for all Boolean algebra. Then in quantum physics, you have two high, but instead of just two points, there’s a whole circle.

### Rickles:

Yeah, [overlapping voices] [?] —

### Finkelstein:

So it’s constant [?].

### Rickles:

Yeah, because you’ve got these extra superposition unless it goes here.

### Finkelstein:

Right. You have all those super… So a point and a point and then you have the continuous circle, so divertingly discrete and horizontally continuous. So is that continuous or discrete? Yes. [Laughs]

### Rickles:

I’m just trying to think in terms of looking at the path interval approach, whether you had any instinct to try and —

### Finkelstein:

As far as I’m concerned, the path interval is just an operated product. If you look at the product of several operators, A, B, C, D, and put the indices in, then one way of looking at that summation is as a sum of all the paths joining the indices. I’m sure that’s how the Feynman path emerges from a more quantum theory — just an operator product. The important thing is that the thing you’re multiplying is the exponential of a classical phase of a classical action that Dirac saw analogy and Feynman saw an exactness but couldn’t express it because of his commitment to the continuum. I’m sure that’s going to survive.

### Rickles:

But you’ll have a different measure, because I’m just wondering if you… There are elements of your work, especially when you get to the space-time code papers, that look very much like causal set theory. I’m just trying to figure out whether there were some causal set kinds of things going on earlier.

### Finkelstein:

That was… what’s the name again?

### Rickles:

Dantzig?

### Finkelstein:

Causal sets.

### Rickles:

Sorkin.

### Finkelstein:

Right, Rafael Sorkin. I think I put him on to causal sets. In doing the homotopy analysis of gravity, I realized that what counted were the light cones, and they in turn just expressed the causal relation. So I defined the manifold by a causal relation and Revol’t Pimenov picked up on this in Russia and wrote a book about…

### Rickles:

Pimenov?

### Finkelstein:

At the time, he was in a concentration camp. He was guilty of self-publication. He was quite famous and got a high position when Stalin died and things liberalized. He refers to my work on causal spaces very generously. I had hoped to meet him sometime but was never fortunate.

### Rickles:

Did you know about van Dantzig’s work on…

### Finkelstein:

Numbered language of science? Oh no. I’m thinking of —

### Rickles:

David van Dantzig. I think that’s his name. He has his paper in the… Do you know the 1955 Berne conference for Einstein? There was this paper by van Dantzig.

### Finkelstein:

Which conference?

### Rickles:

The Berne conference, the Jubilee of Relativity, 1955.

### Finkelstein:

Oh yes!

### Rickles:

There’s a paper by van Dantzig where he has this kind of…

### Finkelstein:

I’m not quite recognizing the name of the town as you say it.

### Rickles:

Switzerland, Berne.

### Finkelstein:

Oh, oh! No, let’s see. No, I was thinking of a Warsaw conference.

### Rickles:

Oh, Jablonna.

### Finkelstein:

Jablonna. Okay. No, I wasn’t at the conference you mentioned, and the name van Dantzig is kind of new to me.

### Rickles:

Hmm. Okay.

### Finkelstein:

Should I look at it?

### Rickles:

Maybe. I can send you a copy.

### Finkelstein:

Oh, please do. What kinds of things did he do?

### Rickles:

Well, almost came up with a similar idea to causal set theory, the idea that you modeled space-time by discrete structure that has a property —

### Finkelstein:

The problem with the theory of causal sets…

### Rickles:

Partial [overlapping voices].

### Finkelstein:

…is that it’s leaving quantization for the end.

### Rickles:

Yeah. [Overlapping voices]

### Finkelstein:

It’s building the theory classically, and it’s true every quantum theory has some kind of classical skeleton, so it’s okay. But you shouldn’t… Surely there must be some important physical effects that you discover only after the quantum theory is turned on.

### Rickles:

Actually, another person who springs to mind talking about that is von Weizsäcker.

### Finkelstein:

I mean if action means phase of the wave function, how can you possibly do dynamics on it before you have wave functions with phases?

### Rickles:

That’s true. Did you have any knowledge of von Weizsäcker’s work then?

### Finkelstein:

Oh, we were very close.

### Rickles:

Long ago or when?

### Finkelstein:

I remember our first meeting, I think. I think he was attracted by my work in quantum logic. Right. Yes. He too was struck by the fact that the Lorentz group is SL(2), which is the group of a two-component complex vector, which means that it’s a quantum logic. And because a two-cone vector, it’s the quantum theory of a binary choice. So both of us arrived at the idea that therefore space-time is a collection of binary decisions. I think he did it before me. I’m sure he did. Where did we meet? I guess he invited me to a conference. He had this annual series, Quantum Theories of Space and Time, and I think I went to all of them or all but one. I was in his home often, and he in mine. I visited him after he had Alzheimer’s and probably one of the last visits to pay respects to him. We were brought together in some of the… Well through him, I met Heinrich Saller.

### Rickles:

Is that Zoller?

### Finkelstein:

Pardon?

### Rickles:

How do you spell that name?

### Finkelstein:

Saller. I guess it’s Heinrich Saller, actually.

### Rickles:

Okay. I don’t know that name.

### Finkelstein:

Who is still alive and well. He retired from the Heisenberg — he worked with Heisenberg and retired from the Heisenberg Institute rather recently. But he now edits the journal I used to edit.

### Rickles:

Yeah. We’ll maybe get to that in a bit. So back to Stevens and black holes.

### Finkelstein:

Right.

### Rickles:

Did Penrose visit Stevens before you — I take it you went to King’s.

### Finkelstein:

The first time I ever saw him was in London.

### Rickles:

At King’s College, was it?

### Finkelstein:

Yeah. We were introduced by Dennis Sciama.

### Rickles:

Yeah. Before that, did Felix Pirani visit you at Stevens? Because obviously Felix was at King’s. I’m wondering if it was Felix that invited you back.

### Finkelstein:

My offhand impression, the best I can remember, is that I met him at King’s.

### Rickles:

Felix at King’s as well. Right.

### Finkelstein:

I’m not sure of that.

### Rickles:

How did you get the invite to King’s? Do you remember?

### Finkelstein:

It wasn’t especially King’s for me. I had met Sciama at the meetings.

### Rickles:

Ah, okay. So Sciama.

### Finkelstein:

And he wanted me to meet Penrose, so he set it up, and it happened to be at King’s.

### Rickles:

So Sciama was at the Stevens meetings.

### Finkelstein:

I think so. How else could… yeah.

### Rickles:

So was Bondi. I think maybe Bondi was [unintelligible].

### Finkelstein:

Let me see. Oh, it’s true. No. I don’t remember Sciama being at a Stevens meeting. We could have met at a relativity meeting.

### Rickles:

So Pirani was at the Stevens meetings. Okay.

### Finkelstein:

I know I had close conversations and friendly conversations with Sciama, but I think it was in Europe or England.

### Rickles:

Oh yes. So you went to Royaumont, you say, as well.

### Finkelstein:

Then after I did my own black hole work, Sciama wanted me to meet Penrose. It was at King’s that I showed Roger the black hole, and he showed me the spin networks. I was really trying to quantize space-time, but from the bottom up. The idea that one could just learn from spin something useful about space-time, the idea of building it from spins, I really learned from him. I’m not sure how long it would have taken me to have the courage to do that.

### Rickles:

Yeah, he didn’t —

### Finkelstein:

So that was an important meeting for me.

### Rickles:

So did he have his manuscripts, the spin networks manuscript by then or was it just ideas?

### Finkelstein:

Off the cuff. Oh, he had had the manuscripts. I’m sure he did. But first, he didn’t call them spin networks; he called them mops.

### Rickles:

Mops.

### Finkelstein:

Right.

### Rickles:

Because they’ve got all the strands coming off them and they intertwine, whatever they are. He called them mops. That’s interesting!

### Finkelstein:

But there was an important exchange interaction. He walked away with a black hole, and I walked away with spin networks. We will probably meet again.

### Rickles:

Yeah. He never really did anything beyond that with his spin networks, I suppose. But you did the black hole.

### Finkelstein:

Oh, it stimulated a lot of people.

### Rickles:

Yeah. Oh yeah.

### Finkelstein:

Well, he did. He made twistors.

### Rickles:

That’s true.

### Finkelstein:

Except that twistors have nonlinear. It means he already gave up quantum theory. He probably didn’t realize he was giving it up.

### Rickles:

So when you were in England, were you visiting King’s as a research visitor or you were just passing through to give a talk?

### Finkelstein:

I was there because I had a fellowship at CERN. I was stationed in Geneva for a year and on a sabbatical from Stevens.

### Rickles:

Right. And the Royaumont must have happened around then as well, so that’s…

### Finkelstein:

No, I think Royaumont I added across the ocean. I think I came to Royaumont from this country. Yeah, I remember that.

### Rickles:

All right, so CERN then. So why CERN? Why did you go to CERN?

### Finkelstein:

As a result of my work on plasmas at NYU, I became interested in Budker’s idea of relativistic pinches. He thought of them not so much with thermonuclear application but as guide fields for table-sized synchrotron… table-sized cosmotron. If you have a current loop of relativistic particles traveling one way and antiparticles traveling the other way so you have a relativistic current, the magnetic field is strong focusing at all points. You don’t even have to have an alternating gradient. It just focuses. The pinch effect focuses anything you put in it. I call it a Budker accelerator, and I wanted to build one, or at least take a step toward building one. You have to begin by designing a field coil that will hold such a relativistic pinch in place. I have a patent on a single turn high field betatron which I call the Megatron because the field is about a mega-gauss. I knew that the accelerator group at CERN would have experts that could help me in that area, and the Ford Foundation funded me for a year at CERN. It was there I met David Spizer, first of all, and…

### Rickles:

Jauch?

### Finkelstein:

And Jauch, exactly, who became interested in the quantum logic approach and wrote on it.

### Rickles:

Yeah. I studied his book as a master’s student, actually. So where did that project go? What happened to it?

### Finkelstein:

Well first, I’m afraid I didn’t enjoy the mathematics of relativistic magnetohydrodynamics that much, and probably if I had tried to solve the problem theoretically I’d still be working on it. So I decided to try and build one and see what happens, and that I did at Stevens with help from Bostick, who showed me how to put together vacuum systems and so on. Then it became clear that the circular machine was unstable. I didn’t push it as hard as I could. It might be possible to stabilize it, but that would be… He had no other project on top of the project, lifelong pursuit. So anyway, that’s what brought me to CERN.

### Rickles:

Okay. I mean you mentioned in —

### Finkelstein:

Then at Yeshiva, I went straight. Namely, I thought a straight pinch would have less of a stability problem. Indeed, the whole thing only lasts for an instant. Maybe you can do what you want to do before it disappears and learn a little bit about relativistic pinches and use that for further work. So I built a linear machine, and that picture you saw was of my linear work at Yeshiva.

### Rickles:

That was at Yeshiva, okay.

### Finkelstein:

I put together a little capacitor back in Richard [?] system.

### Rickles:

Let’s see. You’ll have to tell me what… What’s it called, the exact —?

### Finkelstein:

The thing that… the round one called the Megatron.

### Rickles:

What is the name of the —

### Finkelstein:

Oh, see there were just the control instruments.

### Rickles:

For?

### Finkelstein:

For, or most of it said linear pinch. Weakly relativistic, 1 MEV. My students assembled a capacitor bank at 100,000 volts and built a step-up transformer to charge the water capacitor to a million volts and then discharge that into the pinch.

### Rickles:

Did you discuss any of this with Kruskal? He used to work on this, on plasmas and…

### Finkelstein:

I think…

### Rickles:

I mean did you have any —

### Finkelstein:

Kruskal and I had two major overlaps. First, his work on solitons helped me think about the simplest example of a topologically non-trivial field, which also led to something like solitons. I called mine topological solitons to distinguish them from his where the soliton was held together purely by dynamics. So we had sort of casual interaction because of the similarity of that work. The important thing is that after I found my regularization of the Schwarzschild singularity, at one of the plasma meetings, I showed it to him and he immediately took out an envelope and sketched a regularization of my solution. I had only half of the complete solution, and he had worked it out as a homework exercise in a general relativity course he had taken years ago and never thought it was worth publishing.

### Rickles:

So that presumably was with Wheeler, right? He was with John Wheeler, wasn’t he? Was he a grad student of Wheeler’s?

### Finkelstein:

No, not particularly. No. No, he was quite… He was working for the plasma group for years. Oh, at the time he did his relativity homework, I have no idea who he was studying with.

### Rickles:

Yeah, that’s what I… Because I heard a story that the Kruskal paper with his diagrams —

### Finkelstein:

That came later, namely after I published my paper, which footnotes Kruskal, and the Kruskal solution became known. He still wasn’t publishing, and the way I hear it is that Wheeler finally wrote the paper and got Kruskal to sign it.

### Rickles:

Yeah. That’s what I heard. Yeah. Okay, so you’re in discussions with Kruskal when you were writing the kinks paper. Is that right, or was it later?

### Finkelstein:

Yes.

### Rickles:

So Misner and Kruskal.

### Finkelstein:

Right.

### Rickles:

Okay. And you called your horizon a unidirectional membrane.

### Finkelstein:

Right.

### Rickles:

Do you remember who came up with the name horizon?

### Finkelstein:

Horizon to me sounds like Penrose. I’m pretty sure he was the one who called it horizon.

### Rickles:

So do you remember giving the Schwarzschild solution paper at the Stevens meetings, rehearsing it?

### Finkelstein:

I imagine I did. That’s funny. I actually don’t remember doing that. I’m pretty sure I gave it at Royaumont. That I could check by looking at their volume.

### Rickles:

What do you remember of the relativity seminars?

### Finkelstein:

Rather little, actually. Misner spoke on the problem of the dynamics of general relativity, and particularly the huge number of constraints that come up in the theory. I remember later on he spoke. He gave a general colloquium at Princeton, and Oppenheimer was in the audience. Misner pointed out that in general relativity, the Hamiltonian has to vanish identically because changing time is just a change of coordinates, and Oppenheimer didn’t get it. He asked some not terribly intelligent questions, and finally the thing had to go ahead.

### Rickles:

I’d heard Dirac had a problem with that as well. Was that your recollection?

### Finkelstein:

Not that I — That was the main problem he solved, the problem of constraints.

### Rickles:

Oh no, but I’m thinking of the idea that the Hamiltonian vanishes. He had a problem with the idea that you’d have things commuting with —

### Finkelstein:

He was the first person to deal with it, I think. Actually, I’m not quite sure. It comes up really if you try and quantize gravity. There’s no particular need to do a Hamiltonian theory with the classical theory. I don’t know who tried to quantize gravity first, Bergmann or Dirac.

### Rickles:

Oh, it was Bergmann. Dirac’s was quite late actually, ‘58.

### Finkelstein:

That was my impression also, yes. Thank you. The problem of a Hamiltonian theory with such strong constraints was handled more geometrically by Dirac so it could be understood.

### Rickles:

Yeah. So I mean Dirac was looking at just Lorentz invariant theories when he was doing those lectures on constraints. Okay, so you didn’t have any hand in the running of the relativity seminars.

### Finkelstein:

I believe not.

### Rickles:

It was just all Schiller and Anderson.

### Finkelstein:

Yes.

### Rickles:

So I’m wondering how much of Wheeler’s idea of geons and getting spin-1/2 from funny topological solutions fed into your work, or whether you were already —

### Finkelstein:

Well, he didn’t say topological I don’t think. Geon was just a collection of a light field bound by its gravitational field.

### Rickles:

Oh, but then he sort of had…

### Finkelstein:

Oh, wormholes.

### Rickles:

Yeah, exactly. So it sort of drifted from that into a more general program.

### Finkelstein:

Right.

### Rickles:

The way I saw it, Wheeler was after spin-1/2 basically. So he had Dieter Brill working on trying to get spin-1/2.

### Finkelstein:

See, I didn’t know that. Or maybe I did and I forgot.

### Rickles:

Yeah, I’m curious because it seems like you were sort of doing work that was directly relevant to what Wheeler was after from what I could see.

### Finkelstein:

Right.

### Rickles:

But maybe I’m just curious why Misner didn’t relay that as well. Also, the “Spin without Spin” title, was that yours? Because that sounds quite Wheeler-ish. It’s a Wheelerism.

### Finkelstein:

I don’t know. I don’t think I would have put it that way. That’s a Wheelerism.

### Rickles:

It’s a Wheelerism. Actually, this is from Dieter Brill’s notes. So I had a bunch of his notes where he’d taken titles down, so maybe he’d just come up with that title being a Wheeler student.

### Finkelstein:

Right.

### Rickles:

Do you remember Oskar Klein? He visited Stevens around March 1958.

### Finkelstein:

Who?

### Rickles:

Oskar Klein.

### Finkelstein:

My goodness. I don’t.

### Rickles:

He was talking about five-dimensional relativity and Eddington.

### Finkelstein:

Yes.

### Rickles:

You don’t remember that one. Who else do we have? Arthur Komar? He was at Stevens as well for a bit, right?

### Finkelstein:

I should have mentioned his name. Well, he wasn’t at Stevens.

### Rickles:

Oh, he was Syracuse, wasn’t he? He stayed —

### Finkelstein:

I know him mainly from Yeshiva.

### Rickles:

Ah, he went there. Okay.

### Finkelstein:

We were colleagues there for many years.

### Rickles:

Yes. Did you discuss his… because he had his own particular approach where he was trying to get physical invariant coordinates for general relativity. Did you discuss that approach with him? Did you buy into it?

### Finkelstein:

I’m sure I did. Komar and I spoke quite often. I think his idea of general relativistic Hamiltonian and its dependence on the Observer, I think they’re right on. People looking for the right stress tensor are probably misguided trying to make general relativity too much like special relativity. I think Arthur understood that very well.

### Rickles:

So you didn’t get tempted by any of the approaches, because obviously Bergmann’s school must have been quite a strong —

### Finkelstein:

They’re all working classical space-time, and that’s drowning the baby in the bathwater.

### Rickles:

Yeah. Who else was doing kind of pure approaches to quantum gravity at the time? Or quantum space-time gravity.

### Finkelstein:

By pure you mean quantizing space-time, too.

### Rickles:

Yes.

### Finkelstein:

I’ve tried to trace back the story of quantum space-time significantly and not connected especially through general relativity. Heisenberg was probably the first to suggest it, and his idea was carried by people along paths that were pretty well-defined.

### Rickles:

Yeah. I think now there’s the Snyder paper, right? Hartland Snyder’s paper [overlapping voices].

### Finkelstein:

Wess is a very good man to speak to on this question of how the ideas got from Heisenberg to Snyder to in Russia…

### Rickles:

Dmitri Ivanenko?

### Finkelstein:

Ivanenko, exactly. Yes. And Yang finally because I think he had—

### Rickles:

Oh, he had his lattice thing, didn’t he, a lattice theory? Is that right?

### Finkelstein:

No, no, no.

### Rickles:

I was thinking of Lee, T. D. Lee.

### Finkelstein:

Yang looked at what Snyder had done.

### Rickles:

I didn’t know that.

### Finkelstein:

And it’s either the same year or the next… I think it was a very important paper. Yang published it in 1941, I believe. No, sorry. ‘47. ‘41 was when Feynman tackled quantum space-time and didn’t publish it as far as I know or as far as he knew. The idea of expressing space-time in terms of spins, as far as I know, begins with Feynman. There’s a formula he wrote down while he was still a graduate student before he went to work for Bethe on QED, namely *x* = γ_{1}+γ_{2}. *x* is the sum of a lot of gammas. That’s obviously the Lorentz invariant. He showed me that when I was at Yeshiva. I was sidetracked a little bit by the phenomenon of ball lightning. That came out of the plasma groups.

### Rickles:

I see.

### Finkelstein:

One of the Russians thought that ball lightning might be a natural fusion reactor held together by its own magnetic field. So I generalized the virial theorem to relativistic theory and showed the pressure inside a configuration of plasma and electromagnetic fields couldn’t exceed the pressure outside. There was no significant confinement. I’ve lost my train of thought now.

### Rickles:

We were talking about the development of quantum space-time and then Feynman at Yeshiva.

### Finkelstein:

Quantum space-time, right.

### Rickles:

So Feynman had this approach in his notebook, but he told you about it at Yeshiva. It’s strange how —

### Finkelstein:

I don’t quite see why I drifted off onto ball lightning.

### Rickles:

Because it looked like he had a… Oh, because you were doing that at Yeshiva as well. You got —

### Finkelstein:

Right. Oh yes. I was thinking how I came to be in Feynman’s office. Namely, I went out to measure the conductivity of the rocks beneath the Mohave Desert, and since I was in Pasadena, I just took a chance and went to Feynman’s office. He wasn’t there, but as I was leaving it, he was approaching it. I recognized him and introduced myself, and he said, “Oh, the rabbi,” which is not exactly a good description, but… He associated me with Yeshiva University. He took me to his office, and as much as I was afraid of imposing on his time, he kept me there for hours. It was one of the experiences of my life. I showed him a little bit about quantum logic, which he enjoyed, and he showed me this formula, and he asked me why I think he gave it up. I didn’t have the nerve to say. If I’d been Feynman, I would have said, “You sold out for the Nobel Prize.” I said, “I couldn’t imagine,” and he said, “Because it’s too difficult.” But anyway, that’s how I came to the Feynman formula, and that really affected my subsequent work greatly.

### Rickles:

So is that what you call a Bergmann manifold, this idea that Feynman had?

### Finkelstein:

A what manifold?

### Rickles:

Bergmann manifold.

### Finkelstein:

Why would that be called a Bergmann manifold?

### Rickles:

Well, a manifold with spins.

### Finkelstein:

Oh, no. He didn’t have a manifold that you put spins on. No. His formula is *x* itself is a sum of spin operators.

### Rickles:

Right, okay.

### Finkelstein:

In a Bergmann manifold, you have the same x as in the other manifold, and then you put spins, which are other independent variables, on top of them. This I call a Feynman space-time where the points themselves are assemblies of spins. He did this in 1941. I think I saw that formula in a footnote to one of his papers, but I can’t trace it. He thought so too, and he couldn’t trace it.

### Rickles:

Hmm. I’ll check into it.

### Finkelstein:

He didn’t write it *x* = Σγ. He wrote it as *x* = γ_{1}+γ_{2} and so on. Those ones and twos aren’t the usual subscripts on the γ. Usually it was γ of 1, γ of 2. They’re different spin operators presumably commuting with each other.

### Rickles:

It’s funny. Feynman does seem to have done general relativity throughout but just not published on it. He mentions in various places in his correspondence he’s been working hard on general relativity, but you don’t see it anywhere.

### Finkelstein:

Well, he did publish a little.

### Rickles:

Yeah, in the late ‘50s and early ‘60s, but he’d been working on it for a long time before that, apparently.

### Finkelstein:

Yes.

### Rickles:

I also found in his notebooks with Ted Welton when he was at MIT. They worked through Eddington’s book actually, and they even discuss quantum gravity. Theodore Welton says basically, “Let’s try and do quantum gravity,” and Feynman writes in the margin, “Way too hard,” back then as well. But he kept coming back to it.

### Finkelstein:

Anyway, after Feynman, the next important paper along these lines for me of course is Snyder. Then Yang looks at Snyder, sees that the Lie algebra is not very pretty, and mathematically speaking it’s not simple. I’m not sure it’s even semi-simple. No, it isn’t. It’s singular. The basic quadratic form, the pointing… oh no, the Killing form is singular, and he fixes it. He suggests that a rotation group like SO(5, 1), for example, would have among its generators operators that could serve not only for the operators of the Poincaré group like momentum and rotation, but also the coordinates themselves, the *x*’s, could be rotation operators in this higher dimensional space. So he’s looking now not at a symmetry group so much as a dynamical group of variables. SO(5, 1), he never actually pinned down whether it was SO(5, 1) or SO(3, 3), but some six-dimensional group of we necessarily have an indefinite metric could be what underlies not the Poincaré group but the Heisenberg Poincaré Lie algebra, the algebra you need for a particle in space-time. This was before gauge theory was the rage. Wigner defined a particle as a representation of the Poincaré group, but you can’t do gauge physics if you don’t have an *x* operator, and his particles didn’t have *x*’s. They just had the Poincaré operators. If he’d built an *x*, it wouldn’t be an observable. It wouldn’t even be local. So Yang was really ahead of his time insisting that particles have to have *x*’s, not just the Poincaré group, as if he’s getting ready to do Yang-Mills theory. You can’t do a gauge theory without *x*’s. So Yang put down this rather beautiful theory postulating simple Lie groups. But in my opinion, he lost his nerve halfway through. He represented these groups in the same way Snyder did in Hilbert space whose unitary group is not simple. So on the one hand, he’s looking through into the magic garden of simple groups, but he’s not willing to go there yet. Then later I spoke at his institute at Stony Brook and described his earlier work. None of his students had ever heard of it.

### Rickles:

I can’t believe I’ve never heard of this.

### Finkelstein:

Pardon?

### Rickles:

I can’t believe I’ve not heard of this paper.

### Finkelstein:

It’s incredible.

### Rickles:

I trawled everywhere looking for this type of work. I know Schild’s paper on the discrete approaches, but I’ve never even heard of this. It’s just crazy.

### Finkelstein:

Right. Very few people have. I think it’s a crucial paper. It’s the first simple group. And now if you simply combine Yang and Feynman, instead of representing his group with Hilbert space, you simply represent it with a bunch of spins. The spinning spinors are now spins of SO(5, 1) or SO(3, 3). You put your money on your bet and you take your chances. I waffle. Some days, I use SO(3, 3), and some days I use SO(5, 1). I have a general prejudice of mathematical origin in favor of neutral metrics learned from Cantor, who based his theory of spinors on neutral metrics.

### Rickles:

Cartan.

### Finkelstein:

Cartan. Thank you very much. I keep confusing Cantor and Cartan. They’re almost anagrams. I guess I’m a little bit dyslexic.

### Rickles:

Okay. How are you feeling in terms of… we’ve been going for two hours. That’s pretty good going. I mean I’ve… Where did I want to get to? I’d like to say a bit more about Yeshiva, talk a bit more about Yeshiva, and the space-time code stuff would be good. I don’t know. I’m happy to go that far. If you want to carry on and talk about more things, that’s fine. That’s good.

### Finkelstein:

No, no.

### Rickles:

I don’t want to keep you too long. The ball lightning thing got me curious. How did that actually come about? How did you decide —

### Finkelstein:

Well, it was this Russian proposal that ball — I was in plasma. Ball lightning is a natural thermonuclear reactor maybe, so I had to look into it.

### Rickles:

Oh, so you were sort of officially working on plasma physics.

### Finkelstein:

At Yeshiva at a laboratory.

### Rickles:

Right. Okay. I didn’t know that.

### Finkelstein:

I was trying to make a relativistic pinch.

### Rickles:

Ah, okay. Right. That makes sense now.

### Finkelstein:

A Budker machine, a linear one to begin with.

### Rickles:

Yeah. I noticed you returned…

### Finkelstein:

I think it was at CERN that I… There was one more bit of apprenticeship; namely, there’s the question of the underlying field of quantum theory. If you’re generalizing, you have to consider all the possibilities.

### Rickles:

You mean number field?

### Finkelstein:

Birkhoff and von Neumann had pointed out that R, C and the quaternions are the natural fields for projected geometry. Quaternions are attracted because they’re even more non-commutative, more quantum, than the real and complex quantum theories. So at Geneva, I worked with Spizer and Jauch on quaternion quantum mechanics and immediately ran into the problem that there’s no respectable theory of the tensor product over the quaternion field, which means you can’t do statistics without breaking the symmetry and breaking the field down to the complex or the reals. And you can write Schrödinger’s equation, but that involves singling out an i.

### Rickles:

So what’s the problem with doing tensor product quaternion?

### Finkelstein:

A tensor product of two vector spaces can be expressed in terms of functions which are bilinear, linear in both factors separately. A bilinear function over the quaternions vanishes identically because if you can shoot, you can multiply the first coefficient by i… first space by i, the second by j. Or you can multiply the first by j and the second — You’re supposed to get the same thing either way, and you don’t. If you do, you get zero.

### Rickles:

Yes.

### Finkelstein:

So you have to break the symmetry. You have to single out one quaternion. If you want to preserve the quaternionic symmetry, then you have to introduce into this a new physical variable.

### Rickles:

That will distinguish it.

### Finkelstein:

You could say as an imaginary unit which depends on space and time in the way the gravitational field gives you a local Lorentz group but breaks the global one. So you’d have a local quaternion group, but you’d break the global one. So to gauge theory and this singling out of the i leads to a gauge boson with mass. I’m afraid I’m another one of the infinite number of people who did the Higgs field before Higgs.

### Rickles:

[Laughter] Right!

### Finkelstein:

But I gather the fact that I used quaternion language meant that nobody could read the paper. Those were the good olden days. Now physicists aren’t too well educated in mathematics.

### Rickles:

You didn’t go there to work with Jauch, though, right? You were working on something completely different. Did you know about Jauch?

### Finkelstein:

Did I know about… ?

### Rickles:

About Jauch and Spizer before that?

### Finkelstein:

I didn’t know them before I went to CERN, no.

### Rickles:

So how did you get to working with these guys?

### Finkelstein:

Well, of course I didn’t give up my interests. Youk had written this marvelous book on electrodynamics.

### Rickles:

So he’d already…

### Finkelstein:

I knew of him. I just went out of my way to meet him. I showed him my work on quantum logic, and he was one of the few people who — In fact, he’s probably the only reputable physicist who listened to me. Well, Feynman.

### Rickles:

Yeah, so during that stage —

### Finkelstein:

Feynman came later. Jauch was, at the time, the only one.

### Rickles:

Because especially in CERN, things were going very heavily data-driven, right? So PS had just started.

### Finkelstein:

Well, Jauch wasn’t at CERN so much; he was at the University of Geneva. He was much more theoretical.

### Rickles:

Okay. So do you remember how this kind of work was viewed at the time?

### Finkelstein:

I wasn’t in the particle group. I had virtually no interaction with them. I was often alone.

### Rickles:

Where were you based? Did you have an office over there?

### Finkelstein:

[Overlapping voices] development group.

### Rickles:

And you had like shared office or [overlapping voices]?

### Finkelstein:

I had an office there. I don’t think we shared; I think I had my own office. David Spizer and I would meet there sometimes, and sometimes at the University.

### Rickles:

Were you giving talks or anything at CERN? Was there a kind of research meeting group there at CERN?

### Finkelstein:

I published in the CERN Series. There’s a sequence of three or four papers on quaternion quantum mechanics which are printable in their archives if you need to look them up.

### Rickles:

I got them, I think.

### Finkelstein:

If you do, they’re photographically garbled. It wasn’t a complete set of —

### Rickles:

I think I’m missing two. Or no, maybe —

### Finkelstein:

And the bottom of the pages is missing and so on.

### Rickles:

Yeah, exactly. It’s like quite a multi-part series, right?

### Finkelstein:

It got published later. What’s the name of this Australian philosopher? Hooker.

### Rickles:

Yeah, Cliff Hooker.

### Finkelstein:

Cliff Hooker put the whole quaternion sequence in one of his—

### Rickles:

He had the quantum logic book, yeah.

### Finkelstein:

Right. Do you ever see Cliff Hooker?

### Rickles:

I do, actually. I’ve seen him quite recently. I just did something for a book he edited, in fact, on complex systems.

### Finkelstein:

Give him my regards.

### Rickles:

I will!

### Finkelstein:

I really liked him very much. I’m sorry we’re out of touch.

### Rickles:

I had no idea you knew Cliff Hooker, but yes.

### Finkelstein:

I think we got on well.

### Rickles:

He’s a nice guy. So did this set you back on the path of the logic of quantum mechanics?

### Finkelstein:

The quaternion quantum mechanics helped me because you look at the lattices and so on and you have to make sure. You really have to think about quantum mechanics fresh if you change the underlying number system. Those are very helpful.

### Rickles:

It looks like you were sort of drifting to general relativistic things, and then CERN happens and these papers happen, and then you change direction really is what it looks like from the publication records.

### Finkelstein:

Well, no. I wasn’t drifting toward general relativity. The thing was purely an exercise on the way to quantize space-time. I had hoped to show you couldn’t get spin-½ from general relativity. It turns out you can, but the hell with it. That’s not where it comes from. I mean the real problem is to quantize space-time. It would be nice to know you had to, but you still have to.

### Rickles:

So was there any kind of overlap between Stevens and Yeshiva? It seems like you were… I thought you were sort of —

### Finkelstein:

First, Joel Lebowitz was the first to go from Stevens to Yeshiva. He was the first physicist they hired, and he created the department and was elected head from the very beginning. He hired me from Stevens, and we’re still good friends.

### Rickles:

Was it a teaching institute then? Were you teaching, or was it grad students?

### Finkelstein:

It was the Belfer Graduate School of Science.

### Rickles:

Right. So you were teaching courses to grad students?

### Finkelstein:

Right, and we… No, I taught undergraduates also. That involved crossing a street.

### Rickles:

What was that?

### Finkelstein:

That involved crossing a street.

### Rickles:

Okay, to go to the — [Laughing]

### Finkelstein:

Amsterdam Avenue runs right through the campus, and the religious buildings are on one side, and the School of Science is on the other.

### Rickles:

Yeah. Actually Susskind, in one of his books, paints a nice picture of the early days of Yeshiva as very highly orthodox.

### Finkelstein:

Not highly. Orthodox. But if they were highly orthodox, we wouldn’t have been there. They run the Albert Einstein College of Medicine.

### Rickles:

Susskind says that you arranged his job interview, actually.

### Finkelstein:

I arranged — yeah.

### Rickles:

Is that right? Did you interview him or just you hired him?

### Finkelstein:

Yeah. I was out in California. We met, and he’s obviously promising. We were very lucky that he agreed to come.

### Rickles:

So he must have been working on — Was he working on hadrons? Was he an S-matrix guy? I forget whether he was doing S-matrix stuff. I know he was doing —

### Finkelstein:

I think he was already into string theory.

### Rickles:

I think… No. So string theory… I think he went there in about ‘67, he says, so string theory is…

### Finkelstein:

Sounds late.

### Rickles:

…‘68 you get the dual resonance model stuff, and then ‘69 is when Susskind writes his strings paper. So I’m just wondering what… maybe he was working on hadrons.

### Finkelstein:

Well you know, sometimes people work for years on an idea before they publish. There’s work on the S-matrix which compares the amplitudes for two experiments, which can really be easily related to each other only if you think of them as string interactions.

### Rickles:

Yeah. But the DHS duality, the dual resonance, Rosner-Harari kind of…

### Finkelstein:

Exactly. And he was either doing that when we met or very soon after.

### Rickles:

Okay, so he was already looking along those lines. That’s interesting. Yeah, okay. Because I thought he waited for the Veneziano paper. I know he then started looking at that, but it sounds like maybe he —

### Finkelstein:

It could be. My impression is that he was in that kind of deformation early.

### Rickles:

So other people there, I suppose you had Dirac there for a while. Did —

### Finkelstein:

His office was next to mine.

### Rickles:

Ah, okay. Did you speak?

### Finkelstein:

Once.

### Rickles:

That’s quite a lot compared to most… [laughs].

### Finkelstein:

We just couldn’t stop gabbing.

### Rickles:

So was that about quantum gravity or anything?

### Finkelstein:

Yeah. We quickly came to — He turned the conversation quickly to the question of which is more the fundamental physical constant, e or h. He gave an argument. I guess I knew about it before, and maybe I brought it up because I don’t believe it. He said that e was more fundamental, and his argument is that if h were more fundamental, the formula for e would involve a square root. e is the square root of 137.

### Rickles:

Yeah. That’s a very Eddingtonian…

### Finkelstein:

*hc*, and fundamental laws can’t involve square roots. But from my work in quaternion quantum mechanics, I already knew that e never appears in physics. In fact, in Weyl’s book on electromagnetism, he absorbs an e into the vector potential, so it’s not there in the coupling between the vector potential and the electron. It appears then only in one place: in front of the Maxwell action. Of course it appears there as e^{2}, and physically, how could E appear by itself? You only know about it from forces between two e’s. He heard me out politely and said nevertheless, he still thinks that e is more fundamental. Still he is not going to look for a formula expressing e in terms of h and c, but he’s going to try and express h in terms of e and c. That means he didn’t think quantum theory was fundamental. Beneath it he thinks there’s a more classical kind of theory, and that h, maybe like Einstein thought, represents instabilities in the classical nonlinear field. This is, I think, shown in his book on quantum theory, which is epical, but is not operational. He never says what operations correspond to his state vector any more than von Neumann does. It’s perfectly clear from Heisenberg’s work that the state vector represents an input port or input beam and dual is the output beam, and the basic Born formula is a description of the experiment. So it’s a very pragmatic theory, and anyone who thinks it needs an interpretation just means he hasn’t absorbed its interpretation. But you couldn’t be checking it against experiment as beautifully as we do if it didn’t come with its interpretation. Now any paper on the interpretation of quantum physics is a confession by the author, “I don’t understand quantum theory.” There’s a paper to this effect in Physics Today by a good friend of mine, unfortunately deceased, from Israel.

### Rickles:

Is it a philosopher?

### Finkelstein:

No, a physics major.

### Rickles:

Asher Peres?

### Finkelstein:

Exactly. Thank you. You’re inexhaustible.

### Rickles:

[Laughs] Yeah, I know that paper.

### Finkelstein:

He had a paper saying quantum theory needs no interpretation. He’s right on. Except I don’t think he gives the interpretation. It’s okay; you can’t get everything in one.

### Rickles:

No, but he did have that viewpoint in mind, actually.

### Finkelstein:

Right.

### Rickles:

Yeah. So did you host Penrose as well? And I heard Aharonov was there? Was Aharonov there when you were there?

### Finkelstein:

Penrose may have passed through.

### Rickles:

He wasn’t a visitor.

### Finkelstein:

But he wasn’t on the faculty. And who did you ask about?

### Rickles:

Aharonov. Yakir.

### Finkelstein:

Yes. Aharonov got his first job outside Israel at Yeshiva. Essentially he was sent by Bohm, whom I still respect, and was happy at Yeshiva for quite a few years.

### Rickles:

Actually, on Bohm, did you have any interactions with Bohm?

### Finkelstein:

Oh, very strong.

### Rickles:

Was he a visitor? I mean he had trouble in [overlapping voices].

### Finkelstein:

Not at Yeshiva. But when I spent a sabbatical at Oxford, Bohm would come down from London very often. We’d walk by the river and talk for hours.

### Rickles:

Yeah. I stumbled across a paper of his recently from I think it was late ‘60s where he had an almost causal set kind of approach as well. Have you seen this paper? I should send it to you if you don’t know it because there are some very similar ideas to some of your stuff.

### Finkelstein:

I was especially influenced by his thinking about simplicial complexes. He recognized that the use of the Grassmann algebras were really theories of simplicial complexes. I don’t think it was a very common idea at the time. If you look in Chevalley’s book on modern concepts, he points out that the theory of simplicial complexes is an exercise in Grassmann algebra. What Bohm adds to that is that means it must have to do with Fermi statistics. And he was right on. I once asked him how he could possibly do his work on causal theories of quantum theory and yet have this understanding of quantum theory. He explained he wasn’t trying to get rid of quantum theory; he was trying to show there must be levels beneath it.

### Rickles:

How they merged.

### Finkelstein:

The only catch is then why was he making them look so classical, to look more like a return to old thoughts than pursuing the revolutionary road, and he never answered. But his theory, like all the other attempts to make classical interpretations of Ψ, fail on one question: If Ψ is really an object, the physical thing out there, how come it acts as a statistical distribution? How come it gives probabilities? This is the failure of all those theories to close. That one last link is missing.

### Rickles:

So Susskind mentions he remembers you having a debate with Dirac, actually, at some stage.

### Finkelstein:

I think the one I just mentioned.

### Rickles:

No, it was a different one where you were arguing about the zero point energy of empty space and whether it gravitates, however it should gravitate and how that should be treated. Do you remember having that discussion?

### Finkelstein:

Let me think. I don’t remember that. Gee, if I remembered it, I would cherish it. How did I lose it?

### Rickles:

He mentions this in… gosh, which one is it?

### Finkelstein:

I can certainly see that happening. It seems to me that Dirac’s theory makes that zero point energy very natural. It breaks the symmetry of the electron and positron.

### Rickles:

The time varying…

### Finkelstein:

The vacuum is full of these things, and Heisenberg’s symmetrization is a bit of a gimmick. He’s throwing away ad infinity so you can get any answer you like. Why should there be perfect symmetry between the two? It’s begging the question. It’s just saying let’s set the gravitational field equal to zero; never mind what it really is.

### Rickles:

Yeah. Okay. I think I’m nearly done, actually. The space-time code papers. So how do you view this first space-time code paper? Was it sort of a culmination of…

### Finkelstein:

Well first of all, the point I was trying to get across was the idea of the world as a computer. The more I think about that, that could be taken too shallowly, too simplistically. But in classical physics, you don’t have input/output systems. You talk about the way the world is. In computers, you have an input and you have an output or you don’t have a computer, and in quantum theory, you have an input and an output or you don’t have a quantum theory. So you’re much closer to the computer model than to the old platonic model of what physics is about. I guess my paper on the world as computer is the first to appear in the literature the idea of a quantum computer in a paper I gave on one of those island meetings. Corsica? It’s in a book edited by David Spizer. I suggested that the nature is computer and the individual electron spins were either good spins or the bits, and that’s the idea of quantum memory. But my quantum computer is more quantum than the industrial ones. Namely, the addresses are quantum, too. The usual commercial quantum computers are located at macroscopic…the bits are located at macroscopic positions, the classical addresses.

### Rickles:

In the space-time code paper, though, you don’t have that kind of interpretation of quantum computer, do you?

### Finkelstein:

I think when I wrote that paper… First of all, it was before I met Feynman. So the idea that these abstract spin-1/2 bits were actually spins was not comfortable for me. I like to go slower. That avoids making mistakes, but it avoids a lot of inventions, too. I first had the courage to identify these things with spins after my meeting with Feynman.

### Rickles:

So is that the paper where you get special relativity out, the geometry of special relativity from quantum mechanics?

### Finkelstein:

Getting it out, I mean I knew it — Let’s be realistic.

### Rickles:

Derive.

### Finkelstein:

Maybe building it in is another way to put it.

### Rickles:

Right. I’ll show it — yeah, okay.

### Finkelstein:

I mean a prediction, you can’t predict things you already know.

### Rickles:

[Laughs] Tell that to the string theorists. Actually, did you not —

### Finkelstein:

But I guess I wrote that paper before meeting Weizsäcker. So he and I were both influenced by the recognition that the group of special relativity is SL(2,C), which is the group of a bit.

### Rickles:

Okay. And it was completely independent, though. Yeah. That’s interesting.

### Finkelstein:

He quoted rather often at his talks my statement that he was the first person to recognize this identity, and he would then say that I could say that because I had done it independently.

### Rickles:

[Laughs] Okay. Did you not get tempted by some of the goings on in string theory? Susskind got quite into string theory early on. You had no interest?

### Finkelstein:

First of all, as soon as… I had this horrible two years after I discovered what I called kinks, the solitons of the gravitational field, because they’re finite objects. How could they blow up? It took a ridiculously long time for me to realize that there are still an infinite number of degrees of freedom, that there was still a high frequently spectrum, that you can’t make a physics just of kinks. So there would always be divergences in a continuous field theory. But still for a while, the idea of doing a purely topological physics appealed to me. In the paper by Misner and me on topological conservation laws, a vibrating string was the simplest example of solitons. Sorry. It wasn’t a string; it was a strip.

### Rickles:

Yes, that’s right, and you did a twist. It was like you gave a…

### Finkelstein:

Yeah, right. Yeah. So I was tempted by such ideas, but if you like, the Lie algebras aren’t simple. They violate Yang’s principle.

### Rickles:

So that was — Yeah, because… in fact.

### Finkelstein:

They use Hilbert space. They’re as bad as any other field theory. They’d pretend that there’s this cancellation between the fermion and the boson because they have two infinities. Isn’t infinity minus infinity zero? If you want zero, it’s zero, sure. But that’s the most stupid cancellation I’ve ever seen.

### Rickles:

Were you aware of the mathematical side of string theory? Because the business of the Monster Sporadic Group and Leech lattices and all of that kind of thing was —

### Finkelstein:

I didn’t follow that at all, no. Susskind kept me in touch with Witten’s developing ideas, and they all seemed to be mathematicians’ fantasies. They don’t come out of experiment, and both general relativity and quantum theory of course are thoroughly based on very important experiments. There hasn’t been a case really where a good theory was invented out of nothing. Maybe in his later years Einstein drifted away from experiment, but when he was making his great discoveries, he was thinking very, very hard about solid experiments, even Galileo’s.

### Rickles:

So what’s your view of… Do you follow present-day quantum gravity approaches much?

### Finkelstein:

Most of them use classical space-time or… yeah. It could be lattice or it could be manifold, so I haven’t been following them much. I keep in touch with the… I’m blocking names again. Ashtekar and Smolin.

### Rickles:

The loop approach.

### Finkelstein:

Right. I really can’t find the theory. I’ve tried to get it formulated as a mathematical theory, but it seems to be more of a program. It’s just they have a different program. But loops sound like there’s a continuum down there that they’re using, unless they’ve changed the word loop.

### Rickles:

Yeah, it’s not loops anymore. They have essentially spin networks it’s called, so yeah. So they have SU(2) representations.

### Finkelstein:

Right. I believe the spin network still uses classical addresses, so to speak.

### Rickles:

So the idea is, what they say is — this is Rovelli. He says you have a manifold —

### Finkelstein:

If they really believe that spin network, that means they have a discrete space-time.

### Rickles:

That’s what they think. So they use a manifold as an auxiliary device to set the theory up and then psht.

### Finkelstein:

Nothing up my sleeves.

### Rickles:

Yeah, exactly! [Laughter] Okay, I think we’ll end there. This is excellent. Thank you.