The quantum-field-theory tag has no usage guidance.

**0**

votes

**0**answers

45 views

### Boundary conditions for Klein-Gordon equation [on hold]

Let us consider the Klein-Gordon equation
$$(\Box +m^2)u=0,$$
where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.
...

**6**

votes

**1**answer

292 views

### Why does the Bogolyubov transformation work? - In language of Clifford Algebras?

Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l_{2k}\equiv Cl_{2k}\otimes_{\mathbb{R}}\mathbb{...

**3**

votes

**2**answers

173 views

### Two point function of a free scalar field in Euclidean space-time

This question was previously asked here
http://physics.stackexchange.com/questions/251927/two-point-function-of-a-free-massless-scalar-field-in-euclidean-space-time
though I did not get there an ...

**10**

votes

**3**answers

580 views

### References for Yang-Mills Theory

We are looking to run a working seminar about the Yang-Mills story. We hope that our seminars is of interest to analysts (working with curvatures and Ricci flows on Riemannian manifolds), the ...

**4**

votes

**0**answers

69 views

### Relation of BRST model of equivariant cohomology and BRST cohomology?

I'm now reading Kalkman's paper "BRST model for equivariant cohomology and representation for equivariant Thom class". And I've seen his definition for BRST model is
$B=W(\mathfrak{g})\otimes \...

**9**

votes

**3**answers

357 views

### Braided Hopf algebras and Quantum Field Theories

It is well-known, that there are a lot of applications of classical Hopf algebras in QFT, e.g. Connes-Kreimer renormalization, Birkhoff decomposition, Zimmermann formula, properties of Rota-Baxter ...

**5**

votes

**1**answer

132 views

### Primitive log-divergent graphs and convergence of Feynman amplitudes

To a connected graph $G$, quantum field theory attaches the integral
$$
I_G=\int_{\sigma} \frac{\Omega_G}{\Psi_G^2}
$$ where $N_G$ is the number of edges of the graph, $\sigma$ is the simplex of ...

**6**

votes

**0**answers

97 views

### Extreme unitary minimal models of conformal field theory

Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...

**3**

votes

**2**answers

133 views

### Many-Body Green's Functions for Interacting Systems of Fermions

I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition:
...

**0**

votes

**0**answers

72 views

### Relation between quantum affine algebras and W-algebras

In the paper, there is a concept $qq$-characters for W-algebras $W_{q_1,q_2}$. The usual $q$-character is defined for quantum affine algebra in the paper. A $q$-character can be obtained from a $qq$-...

**0**

votes

**0**answers

63 views

### Looking for A. J. Tolland's exposition of non-renormalizability [duplicate]

In the MathOverFlow thread "Mathematical explanation of the failure to quantize gravity naively" there are references to "A.J. Tolland's very nice exposition of nonrenormalizability". However, Tolland'...

**37**

votes

**4**answers

3k views

### How undecidable is the spectral gap?

Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled Undecidability of the Spectral Gap, there is an extended version on arxiv which is 146 pages long. Here is from the abstract:"Many ...

**7**

votes

**1**answer

136 views

### Fourier series of a Wightman field

From a proof that 2D Wightman CFT leads to a vertex algebra [1]:
Let
$$
Y(a,z):=\frac{1}{(1+z)^{2\Delta_a}}\Phi_a\left(i\frac{1-z}{1+z}\right),\quad\text{with}\quad |z|<1.
$$
Here $\Delta_a\ge 0$ ...

**4**

votes

**1**answer

178 views

### Equation of motion for the Lagrangian $\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$, $G$ is unitary $N \times N$ matrix? [closed]

What is the equation of motion for the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$$where $G$ is an $N \times N$ unitary matrix? Could anyone supply a reference to its ...

**8**

votes

**1**answer

457 views

### Mathematics of Chiral Rings

Let $A$ be a graded vector space, and suppose that two commuting differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex.
We now construct $C(A)$, ...

**5**

votes

**1**answer

132 views

### Generating Functional for the Dirac Field, equivalence of expressions

As with the Klein-Gordon field, we can alternatively derive the Feynman rules with the free Dirac theory by means of a generating functional. In analogy with the scalar field theory where $Z[J]$ is ...

**2**

votes

**0**answers

103 views

### Complex scalar field, computation of propagators, four point function [closed]

This is a followup to my previous question here.
In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can ...

**6**

votes

**1**answer

453 views

### Complex scalar field, generating functional?

The Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^* \phi.$$Can anyone work out or provide me a reference to the computation of the ...

**6**

votes

**1**answer

207 views

### Gauge field quantization, electromagnetism

Classical electromagnetism (with no sources) follows from the actions$$S = \int d^4x\left(-{1\over4}F_{\mu\nu}F^{\mu\nu}\right),\text{ where }F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$The ...

**7**

votes

**1**answer

235 views

### The proof that a vertex algebra can lead to a Wightman QFT

On p. 13 of "Vertex Algebras for Beginners", 2nd edition, Kac writes:
"Under certain assumptions and with certain additional data one may reconstruct the whole QFT from these chiral algebras, but we ...

**11**

votes

**1**answer

248 views

### q-Virasoro and q-Heisenberg algebras

The literature has definitions (seemingly plural, though they might be linked) of a $q$-deformed Virasoro algebra. But is there any link of these to a $q$-deformed Heisenberg algebra? (Classically ...

**13**

votes

**1**answer

221 views

### Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds

Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming ...

**4**

votes

**2**answers

449 views

### What does it mean to take the diagonal of the group $SU(2) \times SU(2) $?

I am reading Witten's paper on topological field theories, in specific the topological twist in page 359. In order to perform the twist he takes the diagonal subgroup of $K = SU(2)_{\text{Right}} \...

**5**

votes

**0**answers

249 views

### Understanding Segal's definition of conformal field theory

I have a fundamental problem in understanding Segal's definition of a conformal field theory:
On the one hand his monoidal CFT-functor is a formalization of the fact that, physically, the integrand ...

**1**

vote

**0**answers

62 views

### anomaly polynomial of generalized Hitchin system

I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theroy can be described in terms of ...

**7**

votes

**0**answers

159 views

### Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? :
$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). $...

**14**

votes

**3**answers

524 views

### Quantum Fields and Infinite Tensor Products

As I understand it, a naive interpretation of the state space of a quantum field theory is an infinite tensor product
$$\otimes_{x\in M} H_x,$$
where $x$ runs over the points of space. This ...

**12**

votes

**2**answers

1k views

### How to learn QFT from mathematical perspective?

I want to learn QFT, because I have heard of its applications in mathematics, I am not interested in scattering cross sections and such. Where can I start to learn? Only books I found are either way ...

**3**

votes

**1**answer

203 views

### Why is an extended T(Q)FT called fully local?

Hopefully this question does not double another. If so, don't bother to close this.
An extended topological quantum field theory is sometimes called, 'fully local".
Why is that? I can imagine that ...

**6**

votes

**1**answer

220 views

### Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...

**4**

votes

**1**answer

299 views

### AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?

**7**

votes

**1**answer

404 views

### TQFT characterization of braiding statistics

In the TQFT language, quasiparticles correspond to Wilson loop operators. It is well-known that quasiparticles can have non-trivial braiding statistics.
Take $2+1$ dimensional Abelian Chern-Simons ...

**1**

vote

**0**answers

93 views

### Dixon's diagram for BRS cohomology

The article by J. A. Dixon titled Calculation of BRS cohomology with spectral sequences (Comm. Math. Phys. Volume 139, Number 3 (1991), pages 495-526) describes in words a diagram that is not printed. ...

**2**

votes

**2**answers

225 views

### Quantum Field theory - integral notation

I have a problem with understanding how the resolution of the identity of an operator is presented in some literature for physicists.
I'm a student of mathematics, and I understand the notion of a ...

**4**

votes

**1**answer

164 views

### reference for higher spin - not gravitational nor stringy

Other than the papers of Berends, Burgers and van Dam, are there any papers that study the general case of deforming a free field theory with higher spin fields to be interactive?

**1**

vote

**0**answers

172 views

### Fourier transform of a matrix represented compact lie group

In physics, I come across this kind of integration (in the nonlinear sigma model):
\begin{equation}
S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}]
\end{equation}
...

**9**

votes

**0**answers

229 views

### Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?

This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...

**8**

votes

**2**answers

456 views

### What is the BRST-anti-BRST formalism?

What is the BRST-anti-BRST formalism?
Is the Sp(2) doublet the ghost, antighost pair?
Introductory accounts of this subject seem to be hard to find. I would appreciate a reference for someone who ...

**11**

votes

**0**answers

388 views

### “extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...

**2**

votes

**0**answers

233 views

### Commutation relations for Dirac and Pauli electron

I hope this question makes sense.
Let $\phi$ be a quantized Dirac spinor with four components $\phi_{\alpha}$, $\alpha=1,2,3,4$. If we denote by $\pi$ the conjugate momentum, then they obey the ...

**4**

votes

**3**answers

246 views

### Reference request for a treatment of Schwinger–Dyson equations

Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog?

**6**

votes

**0**answers

108 views

### How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...

**1**

vote

**0**answers

164 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 ...

**14**

votes

**2**answers

534 views

### Infinite dimensional 2-Hilbert spaces

Is there a definition of an infinite dimensional 2-Hilbert space?
Finite dimensional 2-Hilbert spaces have been discussed by Baez in
http://arxiv.org/abs/q-alg/9609018
In the more recent paper by ...

**1**

vote

**0**answers

290 views

### Feynman integrals in algebraic geometry [closed]

In quantum field theory, multi-loop Feynman integrals are basic ingredients of calculating high order corrections. Recently, I have come across the paper A Feynman integral via higher normal functions....

**1**

vote

**0**answers

80 views

### differential forms in double field theory

In double field theory, there are 'double differential forms' meaning that the standard 1-forms $d x^i$ generate an algebra over functions depending on both of the double coordinates
$x^i$ and $\tilde ...

**6**

votes

**3**answers

864 views

### Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...

**2**

votes

**1**answer

381 views

### Comparison of Different Types of QFT

As far as I can tell, there are a number of major types of quantum field theory. For example,
Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT).
Topological ...

**7**

votes

**2**answers

675 views

### 4d Constructive Quantum Field Theory

As a follow up to my previous question (How does Constructive Quantum Field Theory work?), I was wondering what difficulties physicists have had constructing 4d axiomatic qfts. Why has CQFT's success ...

**15**

votes

**2**answers

1k views

### 2d Ising model in conformal fields theory and statistical mechanics

I am not completely sure that this question is appropriate for this mathematical site. But since in the past I did get on MO couple of times nice answers to rather physical questions, I will try. ...