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323 views

Can auxiliary fields be thought of as Lagrange multipliers?

In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable $$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + ...
6
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3answers
1k views

Interesting topics to research in mathematical physics for undergraduates

I'm planning on getting into research in mathematical physics and was wondering about interesting topics I can get into and possibly make some progress on. I'm particularity fond of abstract algebra ...
6
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4answers
722 views

Is QFT mathematically self-consistent?

After recently going through a short program of self-study in quantum mechanics, I was surprised to find a quote attributed to Feynman essentially saying he was extremely bothered by the computational ...
6
votes
2answers
127 views

Exponential of a differential operator

I have a differential operator $L$, $\displaystyle L = i (t\frac{\partial}{\partial z} - z\frac{\partial}{\partial t})$ I can trivially hit this operator to $x,y,z$ and $t$ as $L x$, $L t$, $L y$, ...
6
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4answers
640 views

Connection between Poisson Brackets and Symplectic Form

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements ...
6
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2answers
882 views

complex numbers in optics

I have recently studied optics. But I feel having missed something important: how can amplitudes of light waves be complex numbers? I suppose this is quite fundamental, but I do not find the answer ...
6
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1answer
235 views

Why is the Hodge dual so essential?

It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric ...
6
votes
1answer
135 views

Topological insulators: why K-theory classification rather than homotopy classification?

I am reading a 2009 paper by Kitaev on K-theory classification of topological insulators. In the 4th page, 1st paragraph in the section "Classification principles", he says, Continuous ...
6
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2answers
230 views

How does the proof of operator commutativity work with non-continuous operators?

In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\{\phi_n\}$, then they commute is given as follows : For any $\phi_n$, $$AB\ \phi_n = a_n\ ...
6
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2answers
148 views

fitting free QFTs into the Haag-Kastler algebraic formulation

Has the free Klein-Gordon quantum field theory been fitted into the Haag-Kastler algebraic framework? (Actually, John Baez told me "yes", and he should know.) If so, can you describe the basic ...
6
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718 views

The derivation of fractional equations

Recently I saw some physical problems that can be modeled by equations with fractional derivatives, and I had some doubts: is it possible to write an action that results in an equation with fractional ...
6
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3answers
173 views

Can eigenstates of a Hilbert space be thought of as delta functions?

Say we have an observable that describes a Hilbert space and that observable acts on state kets. Lets take the position observable for example. Then $\langle y|x\rangle = \delta(y - x)$. But can the ...
6
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1answer
137 views

Physical intuition for deformation quantization of Poisson manifolds

First of all, I know almost nothing about physics. I was reading Kontsevich´s paper on Deformation quantization of Poisson manifolds, however I could not figure out what´s the intuition for such ...
6
votes
2answers
445 views

Vector Potential for Magnetic field when the field is not in simply-connected region

According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some ...
6
votes
2answers
235 views

Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian

This is a detailed question about $U(N)$ intertwiners in LQG, and it comes from the the paper by Freidel and Livine (2011 - archive). It is very specific but related to finding a measure on a quotient ...
6
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1answer
154 views

Are group representations possible when the solution space is not a vector space?

As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector ...
6
votes
1answer
155 views

Einstein's equations as a Dirichlet boundary problem

Can Einstein's equations in vacuum $R_{ab} - \frac{1}{2}Rg_{ab} + \Lambda g_{ab}= 0$ be treated as a Dirichlet problem? I am thinking of something along those lines: Consider a compact manifold $M$ ...
6
votes
1answer
414 views

Why Zeta regularization is not valid for multiple-loops?

Why zeta regularization only valid at one-loop? I mean there are zeta regularizations for multiple zeta sums. Also we could use the zeta regularization iteratively on each variable to obtain finite ...
6
votes
1answer
76 views

Determining the Hodge numbers of some orbifold examples

I'm currently reading about complex geometry in order to get a feeling of how to determine the Hodge numbers, e.g. of certain orbifold constructions. Since I'm a physicist with no deeper mathematical ...
6
votes
2answers
215 views

Ambiguity in number of basis vectors [duplicate]

The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\{|n\rangle\}$ in the problem of particle in a box which ...
6
votes
2answers
151 views

Physical consequences of non-abelian non-trivial holonomy

The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#Significance) can be well described and explained in terms of holonomy of the $U(1)$ connection of the ...
6
votes
2answers
227 views

Is the Green function a prescription for a connection?

I'm trying to learn connection on principal fibre bundle. As far as I can see, the connection is just a given prescription for the displaced field/function on the base space to remains on the ...
6
votes
1answer
209 views

Is the Hilbert space of $\phi^4$ theory known?

Consider free, real scalar field theory in $d=1+3$ dimensions: $H = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi + \frac{1}{2} m^2 \phi^2$. The Hilbert space of this theory is known; it is just ...
6
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2answers
683 views

Book covering Topology required for physics and applications

I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ...
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2answers
848 views

Sources to learn about Greens functions

For a physics major, what are the best books/references on Greens functions for self-studying? My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary ...
6
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2answers
399 views

$\nabla ^2\psi$ equals $\psi -$ average value of $\psi$ at neighboring points

Let $\psi (x,y,z)$ be a scalar field. I found the following statement in Morse & Feshbach Methods of Theoretical Physics: The limiting value of the difference between $\psi$ at a point and the ...
6
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1answer
574 views

CFTs and formalizing quantum field theory

Moshe's recent questions on formalizing quantum field theory and lattices as a definition of field theory remind me of something I occasionally idly wonder about, and maybe this site can tell me the ...
6
votes
1answer
168 views

Can You Obtain New Physics from the use of Fractional Derivatives?

I was curious if anyone could give me an example of the use of fractional derivatives in physics and explain what they offer that "conventional" mathematics does not (in terms of new physics and not ...
6
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1answer
374 views

Nonseparable Hilbert space

What kind of things can go wrong if we try to do quantum mechanics on a nonseparable Hilbert space? I have heard that usual mathematical manipulations that we take for granted will no longer hold. ...
6
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2answers
799 views

A question on the existence of Dirac points in graphene?

As we know, there are two distinct Dirac points for the free electrons in graphene. Which means that the energy spectrum of the 2$\times$2 Hermitian matrix $H(k_x,k_y)$ has two degenerate points $K$ ...
6
votes
1answer
269 views

Thermodynamic limit “vs” the method of steepest descent

Let me use this lecture note as the reference. I would like to know how in the above the expression (14) was obtained from expression (12). In some sense it makes intuitive sense but I would ...
6
votes
1answer
535 views

There seems to be no definition of “stability” in axiomatic QFT. Is there? And, if not, is this a problem?

"stability" is invoked as the justification for the axiomatic requirement that the spectrum of the generators of the translation group must be confined to the forward light-cone. The spectrum ...
6
votes
1answer
94 views

Are observables associated to spacetime regions?

In the Haag-Kastler approach to axiomatic quantum field theory, it is assumed that observables are 'associated' to spacetime regions. What this actually means is that there is a map $\mathcal{A}: R ...
6
votes
1answer
212 views

Equivalent Representations of Clifford Algebra

I'm reviewing David Tong's excellent QFT lecture notes here and am a little confused by something he writes on page 94. We've considered the standard chiral representation of the Clifford Algebra, ...
6
votes
1answer
436 views

Boundary Conditions Invariant Under Conformal Transformations in Electrostatics?

in two dimensional electrostatics it is assumed that the whole physical system is translationally invariant in one direction. Here, the two-dimensional Laplace equation $$\Delta \phi(x,y) = ...
6
votes
2answers
431 views

How do we resolve operator ordering ambiguities when quantizing generic nonlinear second-class constraints?

Dirac came up with a general theory of constraints, including second-class constraints. To quantize such systems, he first computed the Dirac bracket classically, and only then "promoted" the ...
6
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1answer
199 views

Nonextensive statistical mechanics

I know that the Tsallis($S_q$) entropy is called nonextensive information measure in the sense that if $P$ and $Q$ are two probability distributions then $S_q(P\times ...
6
votes
2answers
369 views

Lagrangians combining terms with 1 and 2 derivatives

How are field theory Langrangians treated when some terms have 2 derivatives but others have only 1? Because the number of derivatives in a Lagrangian term is more easily even than odd, the ...
6
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0answers
133 views

What does “mathematically well defined” quantum field theory mean? [duplicate]

Reading Wald's book (page 380, end of the first paragraph of section 14.1) while the author is giving an overall discussion of quantum field theories you can read However, for the more interesting ...
6
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3answers
188 views

Resources showing how to use differential forms in Physics

I've been learning for a while about multivectors and forms and how they simplify many things that in simple vector calculus seems to be complicated. The only problem until now is that differently ...
6
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0answers
238 views

Proof for the completeness of eigenfunctions of a self-adjoint operator

I always heard the eigenfunctions of a self-adjoint operator form a complete basis. Where can I find a proof in infinite dimension space? Presumably readable for physicists.
6
votes
1answer
172 views

Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
6
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0answers
190 views

Prequisites to learn Topological Field Theory? [closed]

Sorry for the somewhat qualitative question but what are the essential prerequisites for someone wanting to learn topological field theory from say the more physical side of things? The math side also ...
6
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0answers
191 views

Physical interpretation: weighted eigenvalues of the Laplacian with a potential

I'm a mathematician with only the basic knowledge of Physics, so my question may be trivial: in this case, mercy me. :-) Let $\Omega \subseteq \mathbb{R}^N$ be a domain and let $V,m:\Omega \to ...
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0answers
40 views

What is the importance of studying degeneration on $M_g$

Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study ...
6
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1answer
348 views

The role of metric in the Wave Equation

The wave equation is often written in the form $$(\partial^2_t-\Delta)u=0,$$ involving the Laplace-Beltrami operator $\Delta$. However, the Laplace-Beltrami operator $\Delta$ is defined only in the ...
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247 views

1-form formulation of quantized electromagnetism

In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps ...
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6answers
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Laplacian of $1/r^2$ (context: electromagnetism and poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is ...
5
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3answers
401 views

What is the purpose of differential form of Gauss Law?

I am learning the differential form of Gauss Law derived from the divergence theorem. $${\rm div}~ \vec{E} =\frac{\rho}{\epsilon_0}.$$ So far in my study of math and physics, the word "differential" ...
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5answers
570 views

What is the meaning of following expresion $C=\frac{\delta Q}{dT}$ mathematicly

Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even wikipedia) writes wrong expressions (from ...