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4
votes
4answers
36 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $\int_{-\infty}^\infty dk_0 ...
1
vote
0answers
52 views

Geometric algebra and quantum field theory

How does the reformulation of QFT with GA look like? I read that GA can be applied to almost every kind of physics, but QFT is rarely mentioned. Is there a lot of research going on in this direction ...
1
vote
1answer
52 views

I need some help understanding the tensor algebra done this problem.

I often see equations rearranged across an equal sign and I have no clue what tricks and reasoning they are using to arrive at these solutions. The only resources I can find on tensor algebra only ...
2
votes
0answers
76 views

Why is Grassman integration so weird?

Why are Grassman integration and differentiation equivalent? The only justification of this definition I have ever scene is "Well, how else could it work?" Indeed, I don't have any other suggestions, ...
6
votes
1answer
181 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
3
votes
2answers
78 views

What's the calculation formula of topological number for mappings of $\pi_{3}(S^2)=\mathbb{Z}$?

It is well-known that, when mapping $|\vec{n}(\vec{x})|=1$, we can use $N=\int{\mathrm{d}x_1\mathrm{d}x_2\vec{n}\cdot(\partial_1\vec{n}\times\partial_2\vec{n})}$ to calculate the topological winding ...
3
votes
3answers
76 views

How can an improper integral have multiple values?

Integrals like this are said to dependend on the contour of integration: $$\int^{\infty}_{-\infty}\frac{x\sin x}{x^2-\sigma^2}dx=\pi e^{i\sigma}\space \mathrm{or}\quad \pi \cos\sigma $$ How is it ...
0
votes
1answer
36 views

Calculating an integral with a branch cut, using some “uniqueness property”

Consider a complex function $$\tilde{f}(z)=z\int_{M}^{\infty}ds' \frac{\rho(s')}{z-s'} \qquad (1)$$ , where $M>0$ and $$\rho(s')=\frac{1}{s'}\sqrt{1-M/s'}.$$ This function is analytic in the ...
2
votes
0answers
28 views

Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
4
votes
1answer
123 views

Homotopy quantum field theories as functors

A homotopy quantum field theory is a symmetric monoidal functor $\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following ...
1
vote
0answers
95 views

Cyclic Properties of the trace in Quantum Field Theory

I'm trying to figure out what's going on in this paper here between lines 10 and 11. But I'll give a brief rundown of what's going on. We are trying to compute the trace of the exponential operator ...
0
votes
0answers
56 views

Self adjointness for functionals

I have posted this question already in the physics forum, but actually nobody could help. I am sorry, this question is related to quantum field theory. The Schrödinger equation of a free scalar field ...
4
votes
0answers
62 views

Mutlivariable integral, How to compute it? [duplicate]

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$\mathcal{Z_{n}}=\int_{-\infty}^{\infty} ...
10
votes
1answer
309 views

Topological Quantum Field theories

I was wondering about the following on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: Why is this so? Secondly, consider the ...
6
votes
0answers
198 views

Multivariable Integral, How to compute it?

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm ...
2
votes
0answers
58 views

Axiom of choice in proof of Wigner's theorem?

In Appendix A of chapter 2 of "The Quantum Theory of Fields," vol. 1, Weinberg presents a proof of Wigner's theorem: given a symmetry transformation $T$ of rays, one can extend this to a symmetry ...
7
votes
0answers
45 views

Cauchy representation and branch point order

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion ...
10
votes
5answers
177 views

Integral representation for $\log$ of operator

How can one prove that $$ (\log\det\cal A=) \operatorname{Tr} \log \cal{A} = \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \operatorname{Tr} e^{-s \mathcal{A}},$$ for a sufficiently well-behaved ...
2
votes
1answer
85 views

Finding a matrix representation for two Grassmann numbers.

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
3
votes
0answers
101 views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk ...
4
votes
3answers
109 views

Show or prove that $\int_{-\infty}^{\infty}\frac{\sin(x)}{x} \mathrm{e}^{i \alpha x} \mathrm{d}x = \pi$

This particular integral rings a bell in our department(Mathematics). It has yet been solved and proved and keeps showing in every third year Complex Analysis Exam. ** An additional condition is that ...
2
votes
2answers
100 views

Function of a differential operator.

Friends of mine who study Quantum Field Theory asked me about the following problem. The task is to simplify the expression $$ f_1(\frac{d}{dx})f_2(x) $$ so that it doesn't contain derivatives, but ...
0
votes
0answers
59 views

First-order partial differential equation from a Feynman diagram.

Can someone please help me solve this first order partial differential equation? $$u \frac{\partial}{\partial u}f(u,t) = \left(\frac{d-6}{2}\right)f(u,t)+\left(\frac{2(d-3)}{s+u}\right)A_{LO}t^{-1-e} ...
4
votes
1answer
89 views

Higher dimensional analogue of the correspondence between $2$d TQFTs and Frobenius algebras?

It's well known that there is an equivalence between the category of $2$-dimensional topological quantum field theories (TQFTs) over $\mathbb{C}$ and commutative Frobenius $\mathbb{C}$-algebras. This ...
1
vote
1answer
84 views

Particles in Free Fields

For the state $\left|\vec{p}\right> = a_{\vec{p}}^{\dagger}\left|0\right>$ we have the energy $H\left|\vec{p}\right>=E_{\vec{p}}\left|\vec{p}\right>$ ...
1
vote
1answer
67 views

Small Lorentz Transformation

This is very simple and I can 50% understand it but would like to properly understand why it is. If we have an infinitesimal Lorentz transformation $\Lambda^\mu _\nu = \delta^\mu _\nu + \omega^\mu ...
8
votes
2answers
404 views

What is a particle mathematically?

In quantum field theory, what is a particle mathematically? How would you explain to someone who kows alot of math but no physics what a particle is? A simple example model would suffice.
10
votes
1answer
199 views

Applications of TQFTs beyond physics

I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some ...
4
votes
1answer
145 views

Graph theory & Feynman integrals

I am attending a course in Graph Theory and I am interested learning something about applications of this subject to Physics, especially I would like to learn something about Feynman integrals. Could ...
0
votes
1answer
52 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
4
votes
0answers
50 views

local gauge invariance of field's homotopy class? Every map $S^2\rightarrow \mathrm{group } G$ is homotopic to a constant map?

In a discussion of a gauge field theory with gauge group $G$, someone says we can use a celebrated result of E. Cartan to show the gauge invariance of matter field's homotopy class. And Cartan's ...
4
votes
0answers
100 views

Dimensional Regularization

I am studying a bit of theoretical physics (QFT and string theory), and I obviously stumbled upon dimensional regularization. I have been told that this technique has in fact a solid mathematical ...
4
votes
2answers
114 views

Geometric meaning of block-diagonalization of a matrix

some times we need to do block-diagonalization in favor of easy computation. For instance, for a matrix like this $$ \begin{bmatrix} A_{11} & A_{12} & A_{13} & 0 & 0 & 0\\ A_{21} ...
2
votes
0answers
114 views

(Topological quantum field theory) identifying objects of cobordism category

I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I ...
2
votes
1answer
40 views

Lagrangian's arguments only up to the first derivative

A question related to a previous question I've asked. I am wondering why in QFT the arguments of the Lagrangian only go up to the first derivative? I remember hearing someone mention that it has to ...
5
votes
2answers
63 views

Why this definition for “symmetry transformation”?

This question concerns section 8.5.1 in these notes: I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total ...
5
votes
1answer
119 views

Degree of maps on the 3-sphere

I am currently in the process of going through Ticciati's Quantum Field Theory for Mathematicians, which states the following (Theorem 13.7.11): "Let $g$ be a differentiable function from $S^3$ to a ...
0
votes
1answer
74 views

Is this Wick rotation correct? (self-intersecting closed contour)

I wonder if what is shown in figure 9.1 here is correct? Doesn't the contour self-intersect, i.e. it's not a simple closed curve hence the Residue theorem shouldn't apply to this closed contour, ...
1
vote
1answer
86 views

Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
0
votes
1answer
74 views

Residue Theorem for a self-intersecting closed curve?

What does the residue theorem say about a closed curve curve as shown in this figure: figure It seems to me that this curve self intersect at origin. It's related to the Wick rotation and I can't ...
2
votes
1answer
182 views

Wick Rotation Contour doesn't seem to be simply connected?

I've seen this (page 112) Wick rotation from several QFT source and all of them explain really bad at what is going on. From Complex analysis I know that for instance if we have an integral from ...
1
vote
0answers
64 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Is the following operations OK (this is related to the Feynman parameter trick)? $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
5
votes
1answer
307 views

Mathematics and Physics prerequisites for mirror symmetry

I am a physics undergrad interested in Mathematical Physics. I am more interested in the mathematical side of things, and interested to solve problems in mathematics inspired by physics maybe with the ...
0
votes
1answer
67 views

Is this OK: $\int_a^b \!\mathrm{d}x \,\,f(x) =^? \int_{\mathrm{i}\,a}^{{\mathrm{i}\,b}} \!\mathrm{d} (\mathrm{-i}y)\,\,f(\mathrm{-i}y).$ Any proof?

This is related to Wick rotation in QFT but it is not exactly it. I'll take a 2-dimensional spacetime to be brief but usually there are more. I've checked with a few functions and with finite ...
4
votes
3answers
219 views

How is this linear 2nd-order ODE solved?

In this article, the authors present the inhomogeneous equation $$\ddot{\phi}_2 + \phi_2 + g_2\phi_1^2 + \omega_1\ddot{\phi}_1 = 0,\tag{11}$$ where $$ \phi_1 = p_1 \cos(\tau + \alpha), \tag{13}$$ ...
4
votes
1answer
130 views

How to do this integral

Prove that $$\int \frac{d^{n}q}{(2\pi)^{n} }\frac{q^{2a}}{(q^{2}+D)^{b}}=D^{-(b-a-n/2)}\frac{\Gamma (b-a-n/2)\Gamma (a+n/2)}{(4\pi )^{n/2}\Gamma (n)\Gamma (n/2)}$$ The angular part is easy to do as ...
2
votes
0answers
68 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
8
votes
1answer
285 views

What is quantum field in terms of mathematics?

I am reading a book on quantum field theory, while I have never been trained as a physicist. I found a big gap in language and have trouble understanding what physicists mean by "quantum field". If I ...
2
votes
0answers
83 views

Wilson lines, boundary condisions, surface defects of TQFTs

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
3
votes
0answers
123 views

Steps toward making the Feynman integral rigorous

I would like to know if there has been any progress recently toward giving the Feynman integral a rigorous definition. I heard that Richard Borcherds is working on giving quantum field theory a ...