## June 3, 2009

### Mathematical Principles

#### Posted by David Corfield

I’ve been reading several works by Ernst Cassirer of late. In his Determinism and Indeterminism in Modern Physics, Yale University Press, 1956, (translation by O. Benfrey of ‘Determinismus und Indeterminismus in der modernen Physik’, 1936) he discusses the multi-levelled nature of physics: laws encompass measurements, and principles encompass laws.

Here in fact we find a basic methodological characteristic common to all genuine statements of principles. Principles do not stand on the same level as laws, for the latter are statements concerning specific concrete phenomena. Principles are not themselves laws, but rules for seeking and finding laws. This heuristic point of view applies to all principles. They set out from the presupposition of certain common determinations valid for all natural phenomena and ask whether in the specialized disciplines one finds something corresponding to these determinations, and how this “something” is to be defined in particular cases.

The power and value of physical principles consists in this capacity for “synopsis,” for a comprehensive view of whole domains of reality… Principles are invariably bold anticipations that justify themselves in what they accomplish by way of construction and inner organization of our total knowledge. They refer not directly to phenomena but to the form of the laws according to which we order these phenomena. A genuine principle, therefore, is not equivalent to a natural law. It is rather the birthplace of natural laws, a matrix as it were, out of which new natural laws may be born again and again. (pp. 52-53)

An example he has in mind is the Principle of Least Action, a principle of venerable age, up and running by the 1740s, which continued to give birth to natural laws after Cassirer had written these words. Indeed the title of Feynman’s 1942 PhD thesis was The Principle of Least Action in Quantum Mechanics.

Would a similar description lead us to describe mathematics as directed by such principles? If so, what are our best candidates? I suppose ‘Everything is a set’ worked for a time, but it seems to have lost its power in recent decades.

Perhaps some of the category theoretic slogans fare better:

• All concepts are Kan extensions.
• Better to work with a nice category of non-nice objects than vice versa.
• Don’t commit evil.
• Study the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond.
Posted at June 3, 2009 10:44 AM UTC

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### Re: Mathematical Principles

A principle from number theory due to Barry Mazur:

Deputize all of algebraic geometry over number fields to serve as vast “unifiers” of constellations of algebraic numbers.

Posted by: David Corfield on June 3, 2009 12:13 PM | Permalink | Reply to this

### Re: Mathematical Principles

There’s one very basic principle that I always point out explicitly to my undergraduate abstract algebra classes, and which I imagine the clientele here take completely for granted:

To understand some class of mathematical objects, one should study the maps (or more generally, “ways of going”) between them.

Posted by: Mark Meckes on June 3, 2009 1:29 PM | Permalink | Reply to this

### Re: Mathematical Principles

Something I learned from John, (but maybe he was quoting somebody else): “Every interesting equality is a lie.”

Posted by: Thomas Hunter on June 3, 2009 2:17 PM | Permalink | Reply to this

### Re: Mathematical Principles

Yes, we should include a ‘categorification’ principle, like

We understand the true nature of a concept the deeper, the more straightforwardly the definition we use to conceive it lends itself to categorification.

Posted by: David Corfield on June 3, 2009 3:57 PM | Permalink | Reply to this

### Re: Mathematical Principles

My favourite principle would have to be

• The empty set should be a special case, not an exception.
Posted by: Toby Bartels on June 3, 2009 5:27 PM | Permalink | Reply to this

### Re: Mathematical Principles

As grad students at Princeton, we played a game working out which topological properties were satisfied by the empty set.
It sharpened the mind as to precisely what the definition said.

Posted by: jim stasheff on June 4, 2009 1:08 PM | Permalink | Reply to this

### Re: Mathematical Principles

Yes. One interesting test case is “connected”, as was discussed in a related form over at the Secret Blogging Seminar some time back. According to one convention (not the union of two nonempty disjoint opens), it obviously is. According to another (having $\pi_0(X)$ a singleton), it’s not. Desired theorems depend crucially on which convention is chosen!

And which one chooses may say something about one’s mathematical “values”. Which do you think a category theorist is likely to pick?

Posted by: Todd Trimble on June 4, 2009 2:43 PM | Permalink | Reply to this

### Re: Mathematical Principles

Which do you think a category theorist is likely to pick?

If you give up, see p. 53 of Lectures on $n$-Categories and Cohomology.

Posted by: David Corfield on June 4, 2009 2:59 PM | Permalink | Reply to this

### Re: Mathematical Principles

see p. 53 of Lectures on n-Categories and Cohomology.

And yet Mike Shulman, who wrote that page, insists that the simplex category should not be augmented, contadicting

True category theorists, like Mac Lane, never leave it out.

The last time that Mike that tried to justify this (see Discussion at the Lab entry linked above), he found that he had only given another argument for augmenting. I rest my case. (^_^)

Posted by: Toby Bartels on June 4, 2009 11:48 PM | Permalink | Reply to this

### Re: Mathematical Principles

The last time that Mike that tried to justify this, he found that he had only given another argument for augmenting.

That is a completely unfair characterization. Here is what I said:

In my experience, in most applications to homotopy theory, the topologists’ Δ is the important object, both conceptually and mathematically, with its augmented version playing at most a technical role occasionally. And while the augmented Δ does have a cute universal property as a monoidal category, the category of simplicial sets (presheaves on the unaugmented Δ) also has a good universal property: it is the classifying topos for linear orders.

Todd replied that simplicial sets classifies “linear orders with distinct top and bottom” while augmented simplicial sets classify “linear orders with top and bottom, not necessarily distinct.” I agreed. If you choose to regard this as an argument in favor of augmenting, be my guest. I would at least point out that linear orders with distinct top and bottom are important; for instance they are the category in which Freyd’s theorem characterizes the unit interval as a coalgebra for the join endofunctor. Regardless, however, this detracts nothing from my first sentence.

I challenge anyone who believes that the simplex category should always be augmented to construct a model structure on the category of augmented simplicial sets that is Quillen equivalent to the usual one and has all of its good properties. (To start with, that would include: cofibrantly generated with explicitly specified generating sets, all objects cofibrant, left and right proper, and cartesian closed monoidal.) And to define an augmented two-sided bar construction. And maybe to define a version of the Joyal model structure for quasi-categories but using augmented simplicial sets. See also this post.

Posted by: Mike Shulman on June 5, 2009 4:59 PM | Permalink | Reply to this

### Re: Mathematical Principles

I agree with Mike. Both the category $\mathbf{D}$ of all finite ordinals and the category $\Delta$ of nonempty finite ordinals are important. The fact that one is a full subcategory of the other seems to me to be mostly a distraction.

For one thing, there’s the point made in the post that Mike linked to in his final sentence. In short: there’s a general machine that takes in any theory of a suitable kind and spits out a category; and when fed the theory of categories, it spits out the category $\Delta$ (not $\mathbf{D}$).

For another, there’s the fact that for any category $\mathcal{E}$ with finite products, $Colax(\mathbf{D}, \mathcal{E}) = \mathcal{E}^{\Delta^{op}}$ where the left-hand side is the category of colax monoidal functors and monoidal transformations. So given either $\mathbf{D}$ or $\Delta$, you’re forced to think about the other.

This fact also gives an alternative view on what a simplicial object is, and means you can easily generalize Segal’s notion of ‘special simplicial object’ or ‘special $\Delta$-space’ ($\approx$ homotopy monoid) to monoidal categories in which the tensor is not cartesian product. For example, it gives you a definition of homotopy DGA.

Posted by: Tom Leinster on June 6, 2009 2:09 AM | Permalink | Reply to this

### Re: Mathematical Principles

That is a completely unfair characterization.

I wasn't being very serious —hence the smiley—, so I didn't really check for fairness. Sorry if you felt insulted.

However, thank you (and Tom) for the list of examples of the importance of $\Delta$ —or the list of exercises to show how to replace $\Delta$ with $D$. This is the sort of thing that I like to do, and maybe it will help me learn to stop worrying and love $\Delta$.

Posted by: Toby Bartels on June 7, 2009 1:30 AM | Permalink | Reply to this

### Re: Mathematical Principles

The $\pi_0$ version is really path connectedness. Moise/Moore had lots of `pathological’ examples to point out such distinctions. e.g. the Warsaw circle (aka the Austin circle).

Posted by: jim stasheff on June 5, 2009 1:40 PM | Permalink | Reply to this

### Re: Mathematical Principles

The $\pi_0$ version is really path-connectedness.

I won’t dispute that, but it should be pointed out that this fails if you naively define “path-connected” to mean “any two points are connected by a path,” as is done in most undergraduate topology courses.

Posted by: Mike Shulman on June 5, 2009 5:07 PM | Permalink | Reply to this

### Re: Mathematical Principles

Then what is the pi_0 definition?

btw, Moise/Moore would distinguish path connected from arc connected

jim

Posted by: jim stasheff on June 5, 2009 9:48 PM | Permalink | Reply to this

### Re: Mathematical Principles

In this definition, a space is path-connected if it has exactly one path-component, but one has to be careful when interpreting the set of path components $\pi_0(X)$. The point is that the empty space has zero path components if you define $\pi_0(X)$ to be the coequalizer (in Set) of the pair of maps

$hom(I, X) \rightrightarrows |X|$

($|X|$ is the underlying set of $X$) where one map is $h \mapsto h(0)$, and the other is $h \mapsto h(1)$. So by that criterion, the empty space is not path-connected! :-)

It’s a little like saying “we don’t consider 1 to be prime”, and thinking of “(path) connected” as a topological analogue of “prime” – can’t be split up. (I’m speaking loosely but suggestively here.)

Getting back to connectedness though, as opposed to path-connectedness: one neat way of saying a space $X$ is connected is to say that the functor

$hom(X, -): Top \to Set$

preserves coproducts. By that criterion, again the empty space is definitely not connected!

Posted by: Todd Trimble on June 5, 2009 11:53 PM | Permalink | Reply to this

### Re: Mathematical Principles

The point is that the empty space has zero path components if you define $\pi_0(X)$ to be…

I’m having trouble thinking of any definition of $\pi_0(X)$ that would give the empty space any number of path components other than zero. (-:

Well, actually, there is one other possibility: if you define $\pi_n(X,x)$, for a based space $(X,x)$, to be the set of based homotopy classes of based maps $(S^n,0) \to (X,x)$, then $\pi_0$ of the empty space can’t even be defined, because you can’t pick a basepoint! (So, of course, really the set of path components of a space is its fundamental 0-groupoid.)

It is very much like the fact that 1 is not prime, though. All other non-primes are not prime because they have too many factors; 1 is not prime because it has too few. Likewise, all other non-(path-)connected spaces are not (path-)connected because they have too many components; the empty space is not (path-)connected because it has too few. One reason we don’t want 1 to be prime is that otherwise unique factorization into primes would fail: $6 = 2\cdot 3 = 1\cdot 2\cdot 3 = 1\cdot 1\cdot 2\cdot 3 = \dots$. Likewise, one reason we don’t want the empty set to be (path-)connected is that otherwise unique decomposition into (path-)connected components would fail: $X = Y\cup Z = \emptyset \cup Y \cup Z = \dots$.

Posted by: Mike Shulman on June 7, 2009 3:17 AM | Permalink | Reply to this

### Re: Mathematical Principles

Mike: right on all counts. But in fact this discussion points up something even more fundamental that probably gets said “wrong” all too often.

We can say that $\pi_0(X)$ is the set of $\sim$-equivalence classes where $x \sim y$ means there is a path from $x$ to $y$. But then what’s an equivalence class?

• (Wrong) The set of all $x, y$ for which $x \sim y$ is an equivalence class of $X$.
• (Right) The equivalence class of an element $x$ is the set $[x] = \{y: x \sim y\}$. The set of equivalence classes of $X$ is the set of equivalence classes of elements $x$ of $X$.

For example, with respect to the unique equivalence relation on the empty set, there are no equivalence classes. Note this was exactly the issue at hand in giving the “right” notion of path-connectedness. It’s “wrong” to define it as saying “any two points are connected by a path”. It’s “right” first to refer to the path-component of a point, and then say there’s only one such path-component.

Lest anyone think this is all absurdly pedantic or that the “misunderstandings” here are harmless: not so! These issues become more telling when one is dealing with “variable” sets or spaces, as in topos theory or fiberwise topology, where it’s important to get it right the first time, or face some unpleasant consequences!

Posted by: Todd Trimble on June 7, 2009 5:19 AM | Permalink | Reply to this

### Re: Mathematical Principles

Regarding the ‘right’ notion of prime, I definitely agree that you want to exclude 1 most of the time. But I recently had the opposite experience. I wanted my primes to remain primes after (ring-theoretic) localization. But a prime $p$ is invertible in $Z[1/p]$ and hence not prime in the usual sense. But it is prime in the sense that every factorization of it has an invertible factor.

Both “exactly one” and “at most one” are useful concepts in mathematics, but with primality and connectedness we have a word for only one of them. Should we have standard terms like pseudo-prime and pseudo-connected? Actually, I’d prefer “strictly prime” for the one that excludes the trivial case and “prime” for the one that includes it, though that would be impossible to get other people to go along with.

Posted by: James on June 7, 2009 5:39 AM | Permalink | Reply to this

### Re: Mathematical Principles

By analogy with subterminal object, I propose the term subprime for a natural number having at most two factors.

Posted by: Mike Shulman on June 7, 2009 6:41 AM | Permalink | Reply to this

### Re: Mathematical Principles

Is this related to the ‘subprime crisis’?

Posted by: John Baez on June 7, 2009 7:45 PM | Permalink | Reply to this

### Re: Mathematical Principles

The empty set should be a special case, not an exception.

Can we see this as a consequence of something more general? Perhaps the principle which has us work with as nice categories as possible. The category of non-empty sets isn’t so pretty.

John sees it as category theoretic issue, when defining the category of simplices:

If you are a true mathematician, you will wonder “why not use the empty set, too?” Generally it’s bad to leave out the empty set. It may seem like “nothing”, but “nothing” is usually very important. Here it corresponds to the “empty simplex”, with no vertices! Topologists often leave this one out, but sometimes regret it later and put it back in (the buzzword is “augmentation”). True category theorists, like Mac Lane, never leave it out.

Posted by: David Corfield on June 4, 2009 2:11 PM | Permalink | Reply to this

### Re: Mathematical Principles

Now I come to think about it, I’ve been writing about such principles for years.

There’s Arnold in Polymathematics telling us how

The informal complexification, quaternionization, symplectization, contactization etc., described below, are acting not on such small things, as points, functions, varieties, categories or functors, but on the whole of mathematics.

The possibility remains that large aspects of a principles could calve from it, leading to fairly precise laws:

The main dream (or conjecture) is that all these trinities are united by some rectangular “commutative diagrams”. I mean the existence of some “functorial” constructions connecting different trinities. (Symplectization, Complexication and Mathematical Trinities, p. 10)

And Atiyah in Mathematics in the 20th Century,

I have said the 21st century might be the era of quantum mathematics or, if you like, of infinite dimensional mathematics. What could this mean? Quantum mathematics could mean, if we get that far, ‘understanding properly the analysis, geometry, topology, algebra of various non-linear function spaces’, and by ‘understanding properly’ I mean understanding it in such a way as to get quite rigorous proofs of all the beautiful things the physicists have been speculating about.

This work requires generalising the duality between position and momentum in classical mechanics:

This replaces a space by its dual space, and in linear theories that duality is just the Fourier transform. But in non-linear theories, how to replace a Fourier transform is one of the big challenges. Large parts of mathematics are concerned with how to generalise dualities in nonlinear situations. Physicists seem to be able to do so in a remarkable way in their string theories and in M-theory…understanding those non-linear dualities does seem to be one of the big challenges of the next century as well. (p. 15)

Posted by: David Corfield on June 4, 2009 10:17 AM | Permalink | Reply to this

### Re: Mathematical Principles

“Pauli’s Exclusion Principle” by Michela Massimi
The Origin and Validation of a Scientific Principle
1.4.2 Ernst Cassirer and the architectonic of scientific knowledge

“Cassirer reinterprets the ‘a priori’ in regulative terms.
The more we strive to isolate the ultimate common element
of all possible forms of scientific experience that may be
the conditions of possibility of any theory, the more we
become aware that at no given stage of knowledge can this
goal be perfectly achieved. Yet, it remains a regulative idea,
prescribing a fixed direction to the evolution of science.
This reinterpretation of the a priori as a regulative idea
finds its natural expression in what Cassirer called the
‘invariants of experience’.

The a priori no longer denotes that which is prior to experience
in the sense of being the condition of possibility of experience;
but rather that which is the ultimate ‘invariant’ of experience,
unattainable at any stage and yet a regulative goal of scientific
inquiry. This seminal investigation – carried out in Substance
and Function –was further articulated and explored in Cassirer’s
later books, namely in the one dedicated to the philosophy of the
Enlightenment, and in Determinism and Indeterminism in Modern Physics.”

Posted by: Stephen Harris on June 4, 2009 12:18 PM | Permalink | Reply to this

### Re: Mathematical Principles

Thanks. I should read this. It sounds as though Massimi is describing fourth level – the general principle of causality. Sitting higher than particular principles, such as those of least action and of conservation of energy, it may be hard to imagine what more there is to say.

What is the significance of the causal principle and what new insight does it add to what we have learned from the ongoing epistemological analysis? What further step is still to be taken and what new insight concerning scientific knowledge is yet to be expected after we have traversed the earlier levels, after we have progressed from statements of the results of measurements to those of the laws, and from these to statements of principles?

I would like to give an answer to this question, which at first sight will perhaps seem paradoxical. There is in fact nothing left over, nothing new in principle to be added to the description of the process of cognition and of the epistemological structure of science. What the casual principle signifies–and this is the thesis I want to explain and establish in the sequel–is not a new insight concerning content, but solely one concerning mmethod. It does not add a single factor homogeneous with the foregoin, which could be placed alongside it as a material supplement. With regard to content it does not go beyond what has already been observed; it only confirms it and confers upon it as it were the epistemological imprimatur. In this sense it belongs, using the language of Kant, to the modal principles; it is a postulate of empirical thought. And this postulate specifies fundamentally nothing more than that the process, which we have sought to describe in detail, is possible without limitation. It does not maintain that the process of translating data of observation into exact statements of the results of measurements, or the process of gathering together the results of measurements into functional equations by means of general principles, is ever complete. What it demands and what it axiomatically presupposes, is only this: that the completion can and must be sought, that the phenomena of nature are not such as to elude or to withstand in principle the possibility of being ordered by the process we have described. We should understand the causal principle in this sense, and in this sense we will submit it to critical examination in the following pages. For us the causal principle belongs to a new type of physical statement, insofar as it is a statement about measurements, laws, and principles. It says that all these can be so related and combined with one another that from this combination there results a system of physical knowledge and not a mere aggregate of isolated observations. (p. 60)

It seems to me that a similar demand for system is equally present in mathematics.

Posted by: David Corfield on June 5, 2009 9:29 AM | Permalink | Reply to this

### Re: Mathematical Principles

“Thanks. I should read this. It sounds as though Massimi is describing fourth level – the general principle of causality. Sitting higher than particular principles, such as those of least action and of conservation of energy, it may be hard to imagine what more there is to say.”

Thank you. Whenever I’m looking for a new idea to explore, I come here to tune into one of your posts. I’m not sure in what sense you intend “hard to imagine what more there is to say”.
I started philosophy with GEB and so have always used AI as my springboard. In one way or another, I think causality is the root problem of philosophy. It seems like I’ve been reading about it for years, starting with Hume and Kant.

CAUSALITY by Judea Pearl Preface

“Ten years ago, when I began writing Probabilistic Reasoning in
Intelligent Systems (1988), I was working within the empiricist
the foundations of human knowledge, whereas causality simply provides
useful ways of abbreviating and organizing intricate patterns of
probabilistic relationships. Today, my view is quite different. I
now take causal relationships to be the fundamental building blocks
both of physical reality and of human understanding of that reality,
and I regard probabilistic relationships as but the surface phenomena
of the causal machinery that underlies and propels our understanding
of the world.”

SH: So this view seems pretty much contrary to the causality role suggested by Cassirer.

http://terpconnect.umd.edu/~mfrisch/Collins-Hall-Paul/
Causation and Counterfactuals (Representation and Mind)
by John Collins (Editor), Ned Hall (Editor), L. A. Paul (Editor)

“A second and quite serious difficulty has recently been pressed by
Adam Elga (2000).3 Elga argues, very persuasively, that Lewis’s
attempt to provide truth-conditions for a non-backtracking reading
of the counterfactual runs afoul of statistical mechanics.”

“More provocatively, one might hold with Hall (“Two Concepts of
Causation”) that there is no way to provide a univocal analysis of
our ordinary notion of causation, and that therefore the best thing
for a philosopher to do is to break it up into two or more distinct
concepts – distinct, at least, in the sense that they deserve

SH: My understanding is that the philosophical analysis of causality had problems with no solution in sight. Then Lewis in 1973 introduced the notion of counterfactuals which has spawned myriads of new speculations and I think this idea happened after Cassirer. I’m not sure that makes a difference to his argument which I haven’t understood the why of it yet.
“I would like to give an answer to this question, which at first sight will perhaps seem paradoxical. There is in fact nothing left over, nothing new in principle to be added to the description of the process of cognition and of the epistemological structure of science.”

I’ve read a fairly new result by Wolpert
which dismantles Laplace’s demon, but this is about determinism rather than causality.
Wolpert:
“This means, in essence, that Laplace was wrong: even if the universe
were a giant clock, he would not have been able to reliably predict the
universe’s future state before it occurred. Viewed differently, Thm. 2
means that regardless of noise levels and the dimensions and other
characteristics of the underlying attractors of the physical dynamics
of various, there cannot be a time-series prediction algorithm [9] that
is always correct in its prediction of the future state of such systems.”

Posted by: Stephen Harris on June 5, 2009 1:27 PM | Permalink | Reply to this

### Re: Mathematical Principles

I believe you were quoting Cassirer,

“I would like to give an answer to this question, which at first sight will perhaps seem paradoxical. There is in fact nothing left over, nothing new in principle to be added to the description of the process of cognition and of the epistemological structure of science. What the casual principle signifies–and this is the thesis I want to explain and establish in the sequel–is not a new insight concerning content, but solely one concerning method.”
—————————–

And I believe this passage from “Constituting Objectivity” pg. 78 further explains what he meant.
——————————–

“It is essential to grasp that it is not the recognition of the semiotic dimension
in itself that leads to a solution for the epistemological problems of dualism. For as long
as symbols are still looked upon as ‘representations of pre-existing predetermined
things’ the problematic dualist pattern is simply reproduced on this level, offering no
intellectual progress whatsoever. What Cassirer proposes is not a semiotic, but a
Copernican turn, because it implies an inversion of the seeming epistemological
priorities. His rigorous analysis reveals that it is impossible to justify a ‘symbol’ by
referring to the designated, allegedly independent pre-existing ‘entity’ and a _bijective relation of ‘cause’ and ‘effect’_. But inversely it is in fact the elementary possibility of
‘symbolic reference’ without which we would not be able to refer to ‘something’ as ‘this definite object’ at all. In the same way as Kant solved the inconsistencies of the dogmatic approaches by renouncing direct ontological hopes and founding knowledge humbly on what is truly ours, Cassirer reiterates and completes this critical movement for the means of constitution themselves. We cannot explain ‘symbolic reference’ with reference to other phenomena, we cannot go beyond or behind it, because it is the necessary condition which enables us in the first place to address something as a phenomenon at all. It is what we have to start with.”

DC wrote: “It sounds as though Massimi is describing fourth level – the general principle of causality.”

SH: The description certainly seems to involve causality.

Posted by: Stephen Harris on June 6, 2009 6:47 AM | Permalink | Reply to this

### Re: Mathematical Principles

I think Cassirer is using ‘causality’ in an extremely Protean way. I don’t have the book to hand, but this ‘general principle’ is some sort of demand simply that functional relations be sought between quantities. This demand plays itself out over the centuries via principles such as that of least action, but no a priori limits should be set as to how future investigators will unfold it.

In our time, it would encourage us to seek a unified quantum gravity.

Posted by: David Corfield on June 8, 2009 11:09 AM | Permalink | Reply to this

### Re: Mathematical Principles

What may be viewed as such a sort of general principle is mentioned in Y.I. Manin’s article on renormalization as part of “… the on–going epistemological shift related to the foundations of mathematics: discrete and finite nowadays often comes from looking at (homotopy types of) continuous and infinite.”

Posted by: Thomas on June 5, 2009 12:58 PM | Permalink | Reply to this

### Witten’s classification of possible theories is related to the cohomology of space-time; Re: Mathematical Principles

“the on–going epistemological shift related to the foundations of mathematics…”

I find this deeply interesting. Yet few of the physicists with whom I regularly communicate face-to-face ever mention this to me when we discuss Renormalization.

Kenneth G. Wilson didn’t even want to talk about Renormalization when I was running the Physics track at a conference a couple of years ago. He just wanted to expound on his sweeping generalizations about reforming public school education. Which is a goal that I share.

Hate to link to Elsevier, so as text:

Physics Letters B
Volume 213, Issue 3, 27 October 1988, Pages 285-290

Why are there two BRST string field theories?
Physics Letters B, Volume 200, Issues 1-2, 7 January 1988, Pages 22-30
Michio Kaku

Abstract
One of the mysteries of string field theory is why there are two distinct BRST field theories. We solve this puzzle by (1) explicitly constructing a geometric vertex that smoothly interpolates between the other theories, (2) proving that this geometric vertex links the other two by a one-dimensional reparametrization, (3) proving that both BRST theories are thus gauge-fixed versions of a higher, geometric field theory. The geometric theory also explains the origin of the baffling four-string interaction, which now emerges as a gauge artifact, the counterpart of the four-fermion instantaneous Coulomb term of QED.

Physics Letters B, Volume 201, Issue 4, 18 February 1988, Pages 454-458
B. Sathiapalan

Abstract
It is shown that while the usual β-functions approach gives interacting equations of motion that are only valid as small perturbations around the free equations, there is a simple modification that allows one to go off the free mass shell. This involves computing correlation functions of vertex operators that are separated by distances on the order of the lattice spacing — which is equivalent to studying the renormalization group equations with a finite cutoff. Applying this technique to the tachyon equation of motion, in the light cone gauge, we reproduce the off-shell three-tachyon aned vector-tachyon-tachyon vertices of (light cone gauge) open string field theory.

Hamiltonian formulation of string field theory
Physics Letters B, Volume 195, Issue 4, 17 September 1987, Pages 541-546
George Siopsis

Abstract
Witten’s string field theory is quantized in the hamiltonian formalism. The constraints are solved and the hamiltonian is expressed in terms of only physical degrees of freedom. Thus, no Faddeev-Popov ghosts are introduced. Instead, the action contains terms of arbitrarily high order in the string functionals. Agreement with the standard results is demonstrated by an explicit calculation of the residues of the first few poles of the four-tachyon tree amplitude.

and finally BRST-symmetries in free string field theory
Physics Letters B, Volume 177, Issue 1, 4 September 1986, Pages 30-34
Hideaki Aoyama

Abstract
A unitary field transformation useful for examination of the BRST-symmetries in free bosonic open-string field theory is introduced. It is similar to Siegel and Zwiebach’s transformation, but leads to a hermitian BRST-charge in contrast to theirs. The transformed BRST-transformation factorizes into two parts, making the gauge-unfixing procedure rather trivial. Both the minimal-type gauge-invariant action and Witten’s action are dealt with.

New candidates for string field theory from the cohomology of C* algebras
Physics Letters B, Volume 179, Issues 1-2, 16 October 1986, Pages 43-46
Louis Crane, Cesar Gómez

Abstract
Candidates for string field theories are constructed from the equivariant cohomology of two C*-algebras. The C*-algebras are constructed in a standard way from two foliations, corresponding to open and closed strings. The open string case is similar to the * operation of Witten. The classification of possible theories is related to the cohomology of space-time and the cohomology of the Virasoro algebra.

Posted by: Jonathan Vos Post on June 7, 2009 6:30 PM | Permalink | Reply to this