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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Self-energy integral at (complex) polar coordinates [duplicate]

You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta ...
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30 views

Existence of a canonical map of quadratic forms

For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has ...
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24 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
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1answer
8 views

Detail of a trick evaluating HQET Self Energy

I have already asked this question in math SE with no luck. I'm reading the HQET book, and I see this in the book Heavy Quark Physics(the book, it is on Page.79). The original integration is ...
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35 views

Linear Algebra: Inverting an induced operator.

Question: Given an invertible linear map $U:V\to V$, consider the induced map $\tilde{U}:\Lambda^k(V)\to \Lambda^k(V)$ given by $$\tilde{U}(v_1\wedge \cdots\wedge v_k):=\sum_{j=1}^kv_1\wedge \cdots ...
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18 views

A question about functional analysis related to an article of B. Kay

I am reading the article of B. Kay, "A uniqueness result for quasi-free KMS states" (you can download it here, I haven't found it elsewhere) and I am struggling to understand the top of p. 1028. The ...
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14 views

Representation of $SL(2,\mathbb{C})$ over Grassmann algebra

I've noticed that when doing the classical Dirac field, sometimes $\psi(x)$ can be treated as a complex-valued spinor field, but when dealing with canonical or path-integral quantization, it should be ...
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21 views

How to integrate with a matrix in the measure?

I've been given the following integral (actually a path integral from quantum field theory). $$ Z(a;N) = \int d^{2n}M.exp(-\frac{1}{2}tr(M^2)-\frac{a}{M}tr(M^4)) $$ where M is a square matrix of size ...
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23 views

Separable states and entanglement

A bipartite pure state $\rho=|\psi\rangle\langle\psi|\in End(H_A\otimes H_B)$ was not entanglement if there were $|\psi_A\rangle\in H_A$ and $|\psi_B\rangle\in H_B$ such that ...
2
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15 views

Entanglement breaking channels

I can't get the proof of Theorem 2 of this. So we have a superoperator as $\Phi\in End(H_B)$ defined using a POVM $\{R_i\}_{i=1}^k$ meaning, every $R_i$ is a positive operator and ...
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1answer
43 views

Coherent states - operator algebra problem with physics motivation

Motivation: I have a mathematical problem motivated by quantum field theory in physics. It should be quite easy to prove, but for some reason I can't do it. Intro: Let there be operators $\hat{a_i}$ ...
3
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75 views

Atiyah-Segal axioms for TQFT [closed]

Could someone explain the importance of the Atiyah-Segal axioms for TQFT? Why is this studied by mathematicians, why is it interesting or useful?
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1answer
48 views

How to prove the Weyl identity?

In the post, there is a formula called Weyl identity: \begin{align} e^{-\frac{k}{n}a_n z^{-n}} e^{\frac{\ell}{n} a_{-n} w^n} = e^{\frac{\ell}{n} a_{-n} w^n} e^{-\frac{k}{n}a_n z^{-n}} ...
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21 views

Depolarizing channel

At the link I don't understand what is happening for $P_{B|A}$ in the solution of exercise 1. I understand everything up to $P_{B|A}$, first of all what is it? Is it the probability of seeing ...
3
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0answers
31 views

A reference containing computational examples for Quantum information

I need a reference, book, lecture note, webpage, or whatever containing a lot of computed examples. What I need is not just something having definitions, statement, proof. I don't need any ...
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1answer
70 views

Extended Topological Quantum Field Theory, (ETQFT) basics ..

What is the functorial (categorical) definition of a TQFT (Topological Quantum Field Theory), which Jacob Lurie "had extended", for his ETQFT ? Actually I just need to know what are basic tools, to ...
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28 views

Multidimensional integral involving a Gaussian function

I am trying to to show: \begin{equation} \int_{\mathbb{R}^{n}}{}d^{n}\vec{x}\frac{\partial{}}{\partial{}x^{i}}\left(B(\vec{x})\exp\left(-\frac{1}{2}\vec{x}^{T}A\vec{x}\right)\right)=0 \end{equation} ...
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1answer
82 views

Cannot solve this integral used in quantum chemistry

I am writing computer code for an implementation of the Hartree Fock algorithm and I am stuck on a certain integral. This is a great walkthrough to get some background : HFTheory Anyway, the set-up ...
2
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1answer
70 views

What is happening in this integration?

I found in Peskin-Schroeder, while reading Quantum Field Theory. the following integration. $$\frac{1}{4\pi^2 r}\int_m^\infty \frac{se^{-sr}}{\sqrt{s^2-m^2}} = e^{-mr}$$ at the limit $r \rightarrow ...
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33 views

Multidimensional gaussian integral with complex elements.

I am trying to find the answer of this Gaussian integral: \begin{equation} \int{d^{N}\alpha \,\,\,\,e^{-\alpha^{\dagger}M\alpha+\alpha^{\dagger}d}} \end{equation} where $\alpha$ is a vector with ...
3
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1answer
48 views

Trouble evaluating spherical Fourier Transform in Quantum Field Theory

(This is purely for personal study - the exercise is 20.2(a) from Lancaster and Blundell (2014), Oxford Uni. Press - an excellent textbook btw.) "Confirm that the Fourier transform of ...
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2answers
289 views

Laymans explanation of the relation between QFT and knot theory

Could someone give an laymans explanation of the relation between QFT and knot theory? What are the central ideas in Wittens work on the Jones polynomial?
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42 views

Inhomogeneous Klein-Gordon equation (support of the solution)

I would like to solve the Klein-Gordon equation $$ \partial_\mu \partial^\mu \phi + m^2\phi = f, \quad m>0 \qquad (*)$$ on flat Minkowski space, where $f$ is some test function (smooth and ...
6
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1answer
81 views

CFT's vs Vertex Operator Algebras

I am trying to clear my ideas about the relation between a Conformal Field Theory (CFT) and a Vertex Operator Algebra (VOA). For me a CFT based on a (complex) vector space $H$ is a projective monoidal ...
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2answers
62 views

Integral with $\delta$ function

I got this integral in a quantum field theory problem: $$ \int\limits_{-\infty}^{+\infty}\!\!\! dp \, \frac{p^2 \delta\left(\sqrt{p^2-m_2^2}+\sqrt{p^2-m_3^2} -m_1\right)}{\sqrt{p^2-m_2^2} ...
2
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0answers
22 views

Reference request for complex scalar field, propagators worked out with path integral approach? [closed]

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 anyone supply me a reference to where the ...
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2answers
92 views

Why do we use square in measuring a qubit with probability?

A superposition is as follow: $$ \vert\psi\vert = \alpha\vert 0\rangle + \beta\vert 1\rangle. $$ When we measure a qubit we get either the result 0, with probability $\vert \alpha\vert^{2},$ or ...
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25 views

Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert space 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 that ...
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56 views

Chiral algebras for dummies

Perhaps the following is better suited to mathoverflow but I ask here first. Despite the apparent good will of researchers writing on chiral algebras I seem to have troubles understanding basic ...
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2answers
70 views

A question concerning the definition of a local function

What is the exact definition of a local function? Is a function said to be local if it depends only on the value of its variable, and a finite number of derivatives of this variable, at a single ...
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47 views

Any reference where the following “partial integration” or “partial residue” concept is defined?

I am studying the on-shell diagrams techniques to compute scattering amplitudes, let's take as a common reference this paper: http://arxiv.org/abs/1212.5605. Trying to put the content of the article ...
3
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1answer
47 views

Bordism between bordisms

I am reading about $2$-categories of bordisms, denoted $Cob_2(n)$. This paper by Lurie (see page 10) states that the objects of such a category are closed $(n-2)$-manifolds and and for $M, N \in ...
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24 views

What is the geometric form E8?

All the information I have is from the TEDtalk (http://www.ted.com/talks/garrett_lisi_on_his_theory_of_everything) The wikipedia page is a bit too complicated for me... Does it "replace" string ...
3
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1answer
121 views

QFT and topology

I have had a course in topology, I have heard of homotopy quantum field theory and topological field theory, but I dont know anything about QFT, what would be a good starting point to learn about the ...
2
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0answers
292 views

Kock Frobenius algebras and 2D TQFTs

My mathematical career consists of selfstudy of the first 7 chapters of Lee topological manifolds. I want to read the book Kock Frobienius algebras and 2d TQFTs. Can you suggest which books and what ...
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70 views

Taking squares or square roots of differential forms?

Reading the recent paper Loop Integrands from the Riemann Sphere by Yvonne Geyer, Lionel Mason, Ricardo Monteiro and Piotr Tourkine I noticed that the authors occasionally seem to take squares and ...
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48 views

Problems 2.3 in Peskin's book

It's about how to evaluate the integral below, $$ D(x-y)=\int \frac{d^3 p}{(2\pi)^3}\frac{1}{2\sqrt{p^2+m^2}}e^{-i\vec p\cdot (\vec x-\vec y)} $$ it describes the amplitude for a scalar particle in ...
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27 views

The amplitude for a photo from a short burst?

I'm imagining a monochromatic point source which is turned on for 1 second. If I take the Feynman propagator $G(x-y) = \frac{1}{(r^2-t^2+i \epsilon)}$ And I integrate it over time for one ...
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27 views

Integrating three propagators at a vertex.

Does this have a solution in terms of known functions: If G(x-y) is the Feynman propagator G(x-y) = 1/ ((x-y)^2 +i epsilon) with x and y as 4D ...
2
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0answers
41 views

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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109 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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164 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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74 views

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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44 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 ...
2
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31 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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52 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 = ...
2
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49 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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25 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 ...
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63 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
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1answer
275 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 ...