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2
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0answers
21 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
0
votes
0answers
31 views

Prove the Wick's Lemma

I'm interested in constructing a nice proof for my thesis of Wick's Lemma. I'm going to state it in a slightly different context (one which is more useful for my purposes). Let $Z_b = ...
0
votes
0answers
36 views

Green's function of Harmonic Oscillator using Fourier modes

First off, I know this is similar to an already answered question concerning the Green's function of a harmonic oscillator. I wanted to ask a question there in the comments, but couldn't due to ...
2
votes
0answers
36 views

A mixed state integrated over Haar measure in QFT

I think that we can write a state in Quantum Mechanics like: $$ \int dx\lambda _x \vert x\rangle\langle x\vert $$ and when we talk about QFT, we would replace integral with functional integrals: $$ ...
0
votes
0answers
17 views

Find the Expectation Value of Basis States

Recently, I have picked up a Quantum Computing class. We have been asked to find the expectation value of $X$ tensor $Z$ and $H$ tensor $H$ $$ (|00>+|11>)/\sqrt(2); \qquad ...
0
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0answers
11 views

Is there a formula for the normal-ordering of a product of creation- and annihilation operators?

Let $\mathcal{H}$ be the 1-particle Hilbert space and $\mathcal{F}=\oplus_{k=0}^\infty\bigwedge^k\mathcal{H}$ the corresponding fermionic Fock space. Then we have the creation- and annihilation ...
1
vote
0answers
50 views

Neveu-Schwarz and Ramond sector in the free fermion CFT

My question is about the Neveu-Schwarz and the Ramond sector in the free fermion CFT. The setup is as follows. We consider two dimensional Minkowski space with a point removed $M = \mathbb{R}^{1, ...
2
votes
1answer
67 views

Represent a square-root of determinant by Grassmann numbers

I am thinking about the representation of $ \sqrt{\det A}$ for a Matrix $A$ . Since it is known that for the Grassmann numbers $ \eta, \eta ' $ the following relation holds: $$ \int \int d \eta d ...
3
votes
0answers
74 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
0
votes
1answer
25 views

Dirac Gamma Matrices identities..

I need your help in order to prove this relation: $$\gamma^{\mu} \not p \gamma_{\mu}$$ that has to give me $$-2 \not p$$ I tried this: $$ \begin{array}{rll} \gamma^{\mu} \not p \gamma_{\mu} & ...
6
votes
1answer
104 views

Dirac Gamma matrix identity

In my library's (old -- 1996) copy of Peskin and Schroeder, there's an identity I'm struggling to prove. In my copy it occurs on page 42, between equations 3.28 and 3.29, but I don't know how well ...
2
votes
1answer
47 views

Dirichlet Series and Asymptotic Expansions

Consider the Dirichlet series $\tilde{f}(s)= \sum_{n=1}^{\infty} f(n) n^{-s}$. In the page "Zeta Function Regularization" of Wikipedia http://en.wikipedia.org/wiki/Zeta_function_regularization I ...
0
votes
2answers
44 views

Help with some notation in QFT

I'm reading a paper on QFT and QEIs but i'm a little sketchy on some of the prerequisites. Can anyone tell me what this represents, $$C_{0}^{\infty}(M)$$ Where M is globally hyperbolic spacetime. I ...
4
votes
3answers
159 views

Replace a sum with an integral $\sum\rightarrow \int$

How can one turn a sum to an integral. Example $$\sum_k f(k) \approx N\cdot\int_k dk\, f(k). $$ How do you find the factor $N$? The quantities should be approximately equal. Example form Peskin ...
1
vote
0answers
103 views

Counting the number of partitions having blocks of cardinality 2 and non-distinct elements

Say I have a set of integers $\{1,2,\cdots,n\}$, then there exists $B_n$ partitions of this set where $B_n$ is a Bell number. For instance, there are $B_3$=5 partitions of the set $\{1,2,3\}$: $$ ...
4
votes
2answers
169 views

question about monoidal structure of a 2-category

Consider an extension of the 1-category of vector spaces and linear maps down to a 2-category $\mathcal{C}$ whose objects are $k$-linear categories. What is the symmetric monoidal structure on the ...
1
vote
0answers
116 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
78 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
1answer
140 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
223 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
92 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
99 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
42 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
1answer
53 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
149 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
170 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
61 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
64 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
397 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
209 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
75 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
55 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
258 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
178 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 ...
4
votes
0answers
193 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
114 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 ...
1
vote
2answers
114 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 ...
4
votes
1answer
97 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
85 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
77 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
420 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.
11
votes
1answer
225 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
181 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
53 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 $S^2$?
5
votes
1answer
61 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 ...
5
votes
0answers
120 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
146 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} ...
3
votes
0answers
127 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
42 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
68 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 ...