DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced ...

learn more… | top users | synonyms

9
votes
0answers
377 views

Noether currents for the BRST tranformation of Yang-Mills fields

The Lagrangian of the Yang-Mills fields is given by $$ \mathcal{L}=-\frac{1}{4}(F^a_{\mu\nu})^2+\bar{\psi}(i\gamma^{\mu} D_{\mu}-m)\psi-\frac{1}{2\xi}(\partial\cdot A^a)^2+ ...
7
votes
1answer
341 views

Are identity types interpreted physically in an infinity-topos formulation of equations of motion?

In reference to Urs Schreibers paper/book on foundations of field theory Differential cohomology in a cohesive infinity-topos I wonder: are identity types there used "only" for the computations, or ...
6
votes
1answer
722 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 ...
13
votes
0answers
379 views

What is Motivic mathematics and how is it used in physics?

In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being ...
12
votes
1answer
288 views

Lie group of Schrodinger Wave equation

In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 ...
9
votes
1answer
176 views

Relation between cohomology and the BRST operator

Given a manifold $M$, we may define the $p$th de Rham cohomology group $H^p(M)$ as the quotient, $$C^p(M) \, / \, Z^p(M)$$ where $C^p$ and $Z^p$ are the groups of closed and exact $p$-forms ...
8
votes
4answers
590 views

Using supersymmetry outside high energy/particle physics

Are there applications of supersymmetry in other branches of physics other than high energy/particle physics?
8
votes
1answer
616 views

Iterated dimensional regularization

Given a 2-loop divergent integral $\int F(q,p)\,\mathrm{d}p\mathrm{d}q$, can it be solved iteratively? I mean I integrate over $p$ keeping $q$ constant Then I integrate over $q$ In both iterated ...
12
votes
3answers
1k views

What are the mathematical problems in introducing Spin 3/2 fermions?

Can the physics complications of introducing spin 3/2 Rarita-Schwinger matter be put in geometric (or other) terms readily accessible to a mathematician?
10
votes
1answer
414 views

False vacuum in axiomatic QFT

There is an elegant way to define the concept of an unstable particle in axiomatic QFT (let's use the Haag-Kastler axioms for definiteness), namely as complex poles in scattering amplitudes. Stable ...
10
votes
3answers
4k views

Integral of the product of three spherical harmonics

Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?: $\int_0^{2\pi}\int_0^\pi ...
8
votes
4answers
632 views

Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where ...
7
votes
1answer
355 views

Wightman axioms and gauge symmetries

I have a basic understanding of the Wightman axioms for QFT. I was reading the about the Mass Gap problem for simple compact gauge groups and was wondering how the gauge group is supposed to be ...
7
votes
2answers
692 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.
7
votes
3answers
2k 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 ...
7
votes
3answers
1k views

Implications of unbounded operators in quantum mechanics

Quantum mechanical observables of a system are represented by self - adjoint operators in a separable complex Hilbert space $\mathcal{H}$. Now I understand a lot of operators ...
5
votes
3answers
275 views

TQFT associates a category to a manifold

Any 3d TQFT (topological-quantum-field-theory) associates a number to a closed oriented 3-manifold, a vector space to a Riemann surface, a category to a circle, and a 2-category to a point. This ...
5
votes
2answers
407 views

Clebsch-Gordan in Fock Space?

When adding the angular momenta of two particles, you use Clebsch-Gordan coefficients, which allow you, in fancy language, to decompose the tensor product of two irreducible representations of the ...
5
votes
3answers
633 views

Axiomatic statistical mechanics

Ive read a few courses on statistical mechanics, and while their textual explanations and example choices differ, the flow of information from microscopy to macroscopy seems the same, and reading ...
4
votes
3answers
371 views

Applying theorem of residues to a correlation function where the Fermi function has no poles

Let $n_F(\omega) = \large \frac{1}{e^{\beta (\omega)} + 1}$ be the Fermi function. A fermionic reservoir correlation function is given by: $$C_{12}(t) = \int_{-\infty}^{+\infty} d\omega~ ...
4
votes
3answers
1k views

Direct Sum of Hilbert spaces

I am a physicist who is not that well-versed in mathematical rigour (a shame, I know! But I'm working on it.) In Wald's book on QFT in Curved spacetimes, I found the following definitions of the ...
4
votes
1answer
902 views

What is a maximal analytic extension?

Can someone explain (as rigorously as possible) what is involved in analytically continuing, say, the Schwarzschild solution to the Kruskal manifold? I understand the two metrics separately but I'm ...
3
votes
2answers
164 views

Symplectic geometry in thermodynamics

There seems to be analogues between Hamiltonian dynamics and thermodynamics given the Legendre transforms between Lagrangian and Hamiltonian functions and all of Maxwell's relations. Poincarè tried to ...
3
votes
3answers
401 views

How to think physically about basic “fields”

"Field" is a name for associating a value with each point in space. This value can be a scalar, vector or tensor etc. I read the wikipedia article and got that much, but then it goes it into more ...
1
vote
1answer
155 views

Calculus of variations and string theory

In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets ...
14
votes
5answers
813 views

Applications of Geometric Topology to Theoretical Physics

Geometric topology is the study of manifolds, maps between manifolds, and embeddings of manifolds in one another. Included in this sub-branch of Pure Mathematics; knot theory, homotopy, manifold ...
8
votes
1answer
645 views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
7
votes
4answers
1k 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 ...
7
votes
2answers
1k 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
votes
2answers
486 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
votes
2answers
207 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 ...
5
votes
1answer
138 views

Transport theorem derivation question

I am trying to rigorously go through some fluid mechanics proofs and theorems. I am currently going through a proof related to the transport theorem and I am having trouble with a step. The steps in ...
4
votes
0answers
170 views

Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
4
votes
1answer
1k views

Local and Global Symmetries

Could somebody point me in the direction of a mathematically rigorous definition local symmetries and global symmetries for a given (classical) field theory? Heuristically I know that global ...
4
votes
1answer
2k views

Uniqueness of Helmholtz decomposition?

Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...
0
votes
1answer
260 views

Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the CF and obtain a summation

Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the correlation function and obtain a summation.
15
votes
2answers
328 views

Geometric quantization of identical particles

Background: It is well known that the quantum mechanics of $n$ identical particles living on $\mathbb{R}^3$ can be obtained from the geometric quantization of the cotangent bundle of the manifold ...
10
votes
1answer
506 views

Intuitively Re-Deriving Equations of Mathematical Physics

Using the intuitive interpretation of the Laplacian $\vec{\nabla}^2$ as the difference between the average value of a field in the neighbourhood of a point & the value of the field at that point, ...
9
votes
1answer
883 views

Quivers in String Theory

Why do a physicist, particularly a string theorist care about Quivers ? Essentially what I'm interested to know is the origin of quivers in string theory and why studying quivers is a natural thing ...
8
votes
2answers
506 views

Is the step of analytic continuation unavoidable or can you model around it?

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure ...
7
votes
1answer
828 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
2answers
174 views

Eigenstates of a Hermitian field operator

Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. ...
6
votes
0answers
179 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
6
votes
1answer
168 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
0answers
257 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
votes
1answer
1k views

What are Wightman fields/functions

Simple question: What are Wightman fields? What are Wightman functions? What are their uses? For example can I use them in operator product expansions? How about in scattering theory?
6
votes
2answers
712 views

What is the symmetry that corresponds to conservation of position?

We know that conserved quantities are associated with certain symmetries. For example conservation of momentum is associated with translational invariance, and conservation of angular momentum is ...
6
votes
1answer
578 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 ...
5
votes
0answers
113 views

Gauge potential for locomotion at low Reynolds number

I've been studying some approaches with gauge theory to some problems in Mechanics and I've found the problem of self propulsion at low Reynolds number a quite complicated one. The approach I'm asking ...
5
votes
1answer
333 views

Grassmann Variables Representation?

It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a ...