In mathematical physics, constructive quantum field theory is the field whose objective is to establish existence theorems for models of quantum field theory. (Def: http://en.m.wikipedia.org/wiki/Constructive_quantum_field_theory)

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Creating an arbitrary state of the quantum simple harmonic oscillator [migrated]

Suppose $\mathcal{B}=\{\lvert 0\rangle, \lvert 1\rangle, \lvert 2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} ...
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calculating Feynman amplitude of a graph

I'm trying to understand Feynman's theorem mentioned in this paper, Chapter 0.0.2. In this paper, the Feynman amplitude of a graph $G$ is a number obtained as a result of the following process: ...
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52 views

Suggest a reading list to start TQFT

What would be books that would give the necessary prerequisities to study TQFT? I want to read something like Kock's Frobenius algebras and 2d TQFTs, I only know enough math that got me through a ...
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113 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 ...
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Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: ...
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39 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 ...
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25 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 ...
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35 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 = ...
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60 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 ...
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39 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: $$ ...
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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 ...
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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 ...
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56 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, ...
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1answer
97 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 ...
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77 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 $$ ...
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1answer
29 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} & ...
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1answer
113 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 ...
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1answer
52 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 ...
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2answers
47 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 ...
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188 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 ...
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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 $\{1,2,3\}$: $$ ...
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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 ...
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127 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 ...
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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 ...
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1answer
148 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, ...
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249 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 ...
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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 ...
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3answers
104 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 ...
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1answer
45 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 ...
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1answer
55 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 ...
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1answer
164 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 ...
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185 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 ...
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64 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 ...
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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} ...
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1answer
417 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 ...
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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 ...
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77 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 ...
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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 ...
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5answers
274 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 ...
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1answer
204 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 ...
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216 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 ...
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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 ...
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2answers
115 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 ...
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102 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 ...
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1answer
86 views

Particles in Free Fields

For the state $\left|\vec{p}\right> = a_{\vec{p}}^{\dagger}\left|0\right>$ we have the energy $H\left|\vec{p}\right>=E_{\vec{p}}\left|\vec{p}\right>$ ...
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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 ...
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429 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.
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237 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 ...
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1answer
192 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 ...
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1answer
56 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 $S^2$?