Questions tagged [quantum-field-theory]

Use this tag for questions about quantum field theory in theoretical/mathematical physics. Quantum Field Theory is the theoretical framework describing the quantization of classical fields allowing a Lorentz-invariant formulation of quantum mechanics. Associate with [tag:mathematical-physics] if necessary.

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Analogues of Wicks/Isserlis theorem to distributions without 4th order interactions?

Suppose I have a zero-centered random variable $x\in \mathcal{D}^n$ where $\mathcal{D}$ is either $\{-1,1\}$ or $\mathbb{R}$, and its density has the following form in Einstein notation convention and ...
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37 views

Integral kernel of $e^{itH_{0}}$

In quantum mechanics, the time-evolved state of a one-particle system is given by $\psi(x,t) = e^{-itH_{0}}\psi_{0}(x)$, where $H_{0} = \frac{1}{2}\Delta$ is the free Hamiltonian, $t\in \mathbb{R}$ ...
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Calculation involving hypergeometric functions

I am trying to derive the normalization constant $N_{k^1,\omega}$ for scalar field modes in an AdS-Rindler wedge. It comes from the following equation: $$\displaystyle \frac{4\pi^2}{\omega N_{k^1,\...
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How to read Alain Connes's Noncommutative Geometry [closed]

I'm a undergraduate student interested in noncommutative geometry and quantum field theory. And I have learned basic functional analysis,theoretical mechanics,basic quantum mechanics, now, I'm ...
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How to deal with contour being “squeezed” by poles?

I'm currently going through this paper and I've run into an issue with a contour integral. The relevant part of section 2.2 is quoted: We now wish to study the analytic structure of (37). In order to ...
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An exponent operator affiliated to a vN algebra.

To begin with I introduce the basic elements of my question. Let $\mathcal{K}$ be a real Hilbert space, for each $f\in \mathcal{K}$ there is a densily defined selfadjoint operator $\varphi(f)$ on a ...
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Berry's curvature vanish in TRS system.

In spin 1/2 system with TR symmetry , the Berry curvature must vanish. Because Berry curvature is odd. How to prove it? \begin{equation} \langle\partial_{-k_x}u^{I}(-k)|\partial_{-k_y}u^{I}(-k)\rangle-...
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$S^3$ hyperspherical harmonics in terms of spherical harmonics

In a problem I might want to work one, I would find myself having to work explicitly with hyperspherical harmonics on a three-sphere and their tensor decomposition. It’s well known that $S^3$ can be ...
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How can I plot this function?

I found this function while looking at quantum field theory. Defined by: $$K(x,y) = \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} e^{i(x k + y q)}\sqrt{ (m^2 + k^2 + q^2) } dk dq$$ ...
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Hamiltonian of the Dirac Field

I was studying Tong's lecture notes and there's a specific mathematical step I do not see how to derive (pages 108, 109); specifically, I do not see how to derive $(5.12)$ Let's go step by step. Given ...
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Plane wave solutions to the Dirac Lagrangian

I was studying Tong's lecture notes and there's a specific mathematical step I do not see how to derive (page 108); specifically, I do not see how to derive $(5.9)$ and $(5.10)$ Let's assume the ...
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I'm having trouble plotting the following qcd function

: $$C(t)=\frac{1}{3}\frac{1}{L^3}\sum_p \frac{p^2}{E_p}e^{-2E_pt}$$ So basically what I'm trying to do is use matlab to plot this function against time. In this case p is 3 momenta vector such that: $$...
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CAR on fermionic Fock spaces

I'm trying to prove by myself the canonical commutation relations (CAR): $$[a(\varphi),a(\psi)] = 0 \quad \mbox{and} \quad [a^{\dagger}(\varphi),a^{\dagger}(\psi)] = 0$$ on fermionic Fock spaces. Here ...
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About sending time to infinity in a slightly imaginary direction in QFT

I am going through the Peskin and Schroeder QFT book. While proving the Gell-Mann and Low theorem in chapter 4 of their book, the authors started with the equation \begin{equation} e^{-iHt}|0\rangle = ...
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How to solve this general gaussian integral without using induction?

The integral which I am talking of can be analytically done by pattern finding and guessing, or by using mathematical induction.This integral arises in calculating the free particle propagator in ...
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Why do we define TQFTs as functors to vector spaces instead of Hilbert spaces?

Let $\mathrm{Cob}_n$ be the category with objects closed oriented $n-1$-manifolds and morphisms being cobordisms identified upto boundary preserving diffeomorphism $\mathrm{Vect}_\mathbb C$ be the ...
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Proving an equation built up out of Dirac-$\gamma$ matrices

Given the following Feynman Amplitude $$\mathscr{M}=\bar{u_s} (\vec p') \Gamma u_r (\vec p) \ \ \ \ (1)$$ Where $\bar u_s, u_r$ are Dirac spinors ($1\times 4$ and $4 \times 1$ matrices respectively)...
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Help with some gamma matrices trace identities

I need to understand the following derivation, but I can't understand which identities it used. Some help or more elaborate derivation will be great
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Understanding the formalism of Spin Sums

I am studying Spin Sums (Quantum Field Theory by Mandl & Shaw, section 8.2) and I have questions about the Mathematics. I'll give some context and then go for the specific questions. The Feynman ...
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Understanding how to get Maxwell's equations in standard form

I am studying Ex.1 of chapter $2$ in Quantum Field Theory's book by Peskin and Schroeder, whose solution is available. Note we're working with Maxwell's equations in vacuum. In section a) we are ...
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How to deal with this Dirac-Delta-function integral

Let me provide you all with some context first. I am studying how to get the differential cross-section formula (in the CoM frame) as explained in Quantum Field Theory's book by Mandl and Shaw (...
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1answer
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Gauss divergence theorem applied to operator valued functions

I'm studing quantum field theory. Especifically the procedure called second quantization for the complex scalar field. I noticed that I can derive the Klein Gordon equation from the Heisemberg ...
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Application of Noether's theorem in Classical Field Theory Peskin's book

I am currently reading Peskin's book about quantum field theory but really struggling with Noether's theorem. I am trying to understand the following example I have given the following Lagrange ...
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Prerequisites for Quantum Fields and Strings: A Course for Mathematicians

I'm interested in learning more about the mathematical structure underneath of quantum field theory and string theory. I've taken a few courses on quantum field theory before, so am getting more ...
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Prove a property of two matrices given three relations

The problem: Given the three relations about square matrices $\alpha$ and $\beta$ $$\alpha^* n \beta^T - \alpha (n+1) \beta^{\dagger} = 1/2$$ $$\alpha^* n \alpha^T + \alpha (n+1) \alpha^{\dagger} = X$...
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Clarification needed for beta function from Ryders book of QFT pahe number 345

Iam studying 1 loop renormalization of QED using QFT by Ryder. In page 345, $e_B=e\mu ^ {\epsilon \over 2} \Bigg(1+{e^2 \over {12\pi ^2 \epsilon}}\Bigg)$ differentiating the above equation gives,...
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Do Eigenvectors of an unbounded operator form a complete basis?

It is common to work with unbounded operators in quantum mechanics and quantum field theory (such as position and momentum operators in QM and field operators and their conjugate momenta in QFT) and ...
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52 views

polylogarithm integrating by parts and expanding

Question: how to go from equation (3) to equation (4)? This is from Horatio Nastase "Intro to Quantum Field Theory" book (Cambridge University Press, 2019) , chapter 59. The reader is supposed to ...
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What are some of the prerequisites for studying quantum field theory? [duplicate]

What are the mathematical and physics prerequisites for studying quantum field theory?
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What are some good introductory books to QFT?

What are some good introductory books to the mathematics of quantum field theory? Basically talks about how exactly it differs from 1st quantisation quantum mechanics.
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Dirac delta multiple integral bounds

I'm struggling to understand the bounds on integration region after performing integral over Delta function. Correct result from book: $$ \int_0^1dz \int_0^1 dy \int_0^1 dx \delta(x+y+z-1) = \int_0^...
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Quantum Computation and Quantum Information: Exercise 2.2

The problem is, Suppose $V$ is a vector space with basis vectors $|0\rangle$ and $|1\rangle$, and $A$ is a linear operator from $V$ to $V$ such that $A|0\rangle = |1\rangle$ and $A|1\rangle = |...
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Can I bring this integral into a form involving known functions?

In the context of quantum field theory, I am facing the following $1$-dimensional integral over Feynman parameters: $$I(a,b) = \int_0^\infty d\alpha \frac{1}{F G} \arctan \frac{F}{G} \tag{1}$$ with ...
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How can I compute this difficult definite integral with logarithm and rational function?

In the context of quantum field theory, I would like to compute the following integral: $$I(a,b) = \int_0^\infty d\gamma \frac{\log (2 a \gamma + b)}{(\gamma + Q + a)(\gamma - Q + a)} \tag{1}$$ with ...
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163 views

Multivariate normal distribution moments

I would like to evaluate the following higher order moments of a multivariate normal distribution in the case of mean $0$ and in the case of mean $\mu$: \begin{equation} E[X_i^{2 n}] \qquad E[(X_i^2 ...
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Reference Request for Quantum Hall Effect

Are there are any good references on the Quantum Hall Effect written for mathematicians? It would be best if it was self-contained, i.e., missing proofs would be OK if they were standard in graduate-...
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Reference moments of non-central multivariate gaussian

In this post they give a nice proof of the Wick theorem which gives a practical way of evaluating the integral \begin{equation} \int x^{k_1} \ldots x^{k_{2 N}} e^{\frac{1}{2} \sum_{i,j} A_{i j}x_i x_j}...
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Integration of a particular quartic form

I would like to solve the following integral: \begin{equation} \int \prod_i d x_i e^{a x_i^2 + b x_i^4 + c x_i^2 x^2_{i+1}} = \int d^n x \;\; e^{(x^2)^T \, A \, (x^2) + B^T (x^2)} \end{equation} ...
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Quantum field theory, interpretation of commutation relation

Let $\phi$ be the quantum field $$ \phi(x) = \int \frac{d^3\mathbf{p}}{(2\pi)^3} \frac{1}{\sqrt{2E_\mathbf{p}}} \Big[ b_\mathbf{p}e^{-ip\cdot x} + c_\mathbf{p}^\dagger e^{ip\cdot x} \Big] $$ with ...
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Witten's description of WZW conformal blocks

I am reading this paper by Witten - Geometric Langlands From Six Dimensions. In section 4.1, he gives a description of the vector space of conformal blocks of the current algebra associated to a ...
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n-dimensional Gaussian integral with absolute values

Given two matrix $A$ and $D$ and a column vector $x$, what is the value of the following integral? $\int d^nx \; \; e^{x^T A x + \mid x \mid^T D \mid x \mid + B x}$ where $\mid x \mid_i = \mid x_i \...
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68 views

Solutions to linear matrix stochastic differential equation

Let A(t), $0\leq t\leq$ T be a random process taking on a value of N$\times$ N real matrices, consider the random matrices Q(t) that satisfy the equation $$\partial_tQ = QA, Q(0) = \hat1$$ then the ...
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Entropies in Algebraic Quantum Field Theory (AQFT)

In finite dimensional quantum information or quantum mechanics. The von Neumann entropy, conditional entropy and mutual information can be defined in terms of the relative entropy as $$H(A)_{\rho} :=...
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Why does the divergence in my integral has a change of type?

I am looking at the following integral in Euclidean space: $$I = \int_{\mathbb{R}} d\tau_3 \int_{\mathbb{R}} d\tau_4 \int_{\mathbb{R}^4} d^4 x_5 \frac{1}{x_{15}^2 x_{25}^2 x_{35}^2 x_{45}^2} \tag{1}$$...
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Rotation in-variance in d=2+1 dimensions (cherns-simons term).

this is probably a stupid question, but, does rotational invariance in $d=2+1$ mean to only rotate the spatial coordinates and not the time. I mean bascially I want to show that $ \int d^3 x \...
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Re-writing Action on a different slice of space-time, Quantum Hall Effect

I'm looking at QHE notes D.Tong and wondering how he gets from equation 5.46 to 5.48 ( http://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf ) $S_{CS}=\frac{k}{4\pi}\int d^3 x \epsilon^{\mu \nu \rho} tr(...
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QHE: effective action as a 'local functional'

' Finally, if we care only about long distances, the effective action should be a local functional, meaning that we can write is as $S_{eff}[A]=\int d^d x... ' Where does this come from and what does ...
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Some questions on exterior algebra

Let $V$ be a finite dimensional vector space over $\mathbb{C}$ and let us assume $\mathcal{V} = \{\psi_{1},...,\psi_{n}\}$ is a basis for $V$. The Exterior (or Grassmann) algebra generated by $V$ is $\...
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Book recommendation for beginners who are physics students but aligned to mathematics

I am a 4th-year undergraduate student and I have fully read R. Shankar's book on Quantum Mechanics and Griffiths book Quantum Mechanics. I have also done a bit of the Application of QM on ...
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Is it okay to do a partial fraction decomposition in that way?

I am facing integrals of the form $$I=\int d^{2\omega} x \frac{1}{(x-x_1)^{2a_1}}\frac{1}{(x-x_2)^{2a_2}} ... \frac{1}{(x-x_n)^{2a_n}}\tag{1}$$ where the $x$’s are $d$-dimensional vectors in the ...

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