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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Quantum mechanical books for mathematicians

I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential ...
user288972's user avatar
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47 votes
3 answers
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reference for multidimensional gaussian integral

I was reading on Wikipedia in this article about the n-dimensional and functional generalization of the Gaussian integral. In particular, I would like to understand how the following equations are ...
harlekin's user avatar
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28 votes
4 answers
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Mathematical and Theoretical Physics Books

Which are the good introductory books on modern mathematical physics? Which are the good advanced books? I read Whittaker's Analytical Dynamics, and I am reading Arnold's Mathematical Methods of ...
superAnnoyingUser's user avatar
27 votes
2 answers
8k views

Relation between $SU(4)$ and $SO(6)$

This is more of a particle physics question than maths. Since $\operatorname{SO}(6)$ and $\operatorname{SU}(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ ...
the lone alien's user avatar
24 votes
2 answers
4k views

What is the relation between representations of Lie Groups and Lie Algebras?

If $G$ is a Lie Group, a representation of $G$ is a pair $(\rho,V)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a group homomorphism. Similarly, if $\mathfrak{g}$ is a Lie Algebra, a ...
Gold's user avatar
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10 votes
0 answers
369 views

Multivariable Integral, How to compute it?

Q 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 f}\left(x_{1},x_{2},\...
Sijo Joseph's user avatar
5 votes
2 answers
763 views

Stueckelberg Feynman propagator computation

On page 35 of Itzykson-Zuber's textbook on quantum field theory, I am having trouble deriving equation (1-180): $\displaystyle G_F(0,r) = \frac{i}{(2\pi)^2 r} \int_m^\infty dp \frac{p}{\sqrt{p^2-m^2}}...
John Jiang's user avatar
3 votes
1 answer
187 views

Show that $\partial_\mu\phi^\ast A^\mu\phi- A_\mu\phi^\ast\partial^\mu\phi=A^\mu\phi\partial_\mu\phi^\ast - A^\mu\phi^\ast\partial_\mu\phi$

The following is loosely related to this question: [...], the most general renormalisable Lagrangian that is invariant under both Lorentz transformations and gauge transformations is $$\mathcal{L}=-\...
Sirius Black's user avatar
2 votes
1 answer
3k views

Prerequisites for ‘Quantum field theory and representation theory: a sketch’ [arXiv:hep-th/0206135]

I'm interested in reading Dr. Peter Woit's article, Quantum field theory and representation theory: a sketch [hep-th/0206135]. What math and physics background would be needed? (A list of topics ...
Sadiq Ahmed's user avatar
79 votes
3 answers
18k views

String Theory: What to do?

This is going to be a relatively broad/open-ended question, so I apologize before hand if it is the wrong place to ask this. Anyways, I'm currently a 3rd year undergraduate starting to more seriously ...
19 votes
1 answer
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Electrodynamics in general spacetime

Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and ...
Daniel Robert-Nicoud's user avatar
17 votes
2 answers
1k 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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16 votes
6 answers
2k views

Products of distributions in QFT

In Quantum Field Theory quantum fields are operator valued distributions. Namely, given the Schwartz space $\mathcal{S}(M)$ defined on Minkowski spacetime $M$, fields are continuous linear maps $\phi :...
Gold's user avatar
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16 votes
1 answer
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Guide to mathematical physics?

I am currently a math phd student specializing in algebraic geometry aspiring to work at the boundaries of the the fields of mathematics and physics and so, was looking into the field of mathematical ...
Sky123's user avatar
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10 votes
2 answers
431 views

Witten's proof of Wick Formula of QFT

Let $\mathcal{S}$ be a finite dimensional real vector space with a positive definite summetric bilinear form $B$. Let $dv$ be a Lebesgue measure on $\mathcal{S}$ such that $$\int_{\mathcal{S}}e^{-B(v,...
Integral fan's user avatar
9 votes
1 answer
1k 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 ...
S.M.Quantum's user avatar
9 votes
4 answers
5k views

Gentle introduction to fibre bundles and gauge connections

To better understand papers like this for example, which makes heavy use of fibre bundles and gauge connections to represent gauge fields, I am looking for a nice introduction to this topic. The only ...
Dilaton's user avatar
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7 votes
2 answers
2k views

Is there some approach to make functional integrals rigorous?

Quantum Mechanics and Quantum Field Theory can both be formulated in terms of the so-called functional integrals. The point is that intuitively it is an "integral over all possible paths" or rather "...
Gold's user avatar
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5 votes
1 answer
514 views

A Grassmann-Variable Identity from Wikipedia

I found this identity on Wikipedia: $$\int\exp\left[\theta^T A\eta+\theta^T J+K^T\eta\right]d\theta d\eta =\det A\exp\left[-K^TA^{-1}J\right]$$ where the integration variables are Grassmann variable....
Jessica's user avatar
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5 votes
1 answer
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What is the correct sign for the four-vector potential gauge transform; $A_\mu\to A_\mu\pm\partial_\mu\lambda$ and where does this gauge originate? [closed]

I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. The convention followed for the Minkowski metric in these ...
Sirius Black's user avatar
5 votes
3 answers
275 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 $$ \phi_1 = p_1 \cos(\tau + \alpha), \tag{13}$$ ...
user avatar
5 votes
1 answer
418 views

Covariant derivative: QFT vs. Math

In class, we have seen that the covariant derivative of some form $R$ can be written as: $$DR = dR + [A, R] = dR + A\wedge R - R\wedge A \tag1$$ Here, $d$ represents the external derivative over ...
Vicky's user avatar
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4 votes
2 answers
1k views

Laplace transform of a product of Modified Bessel Functions

Working with a scalar field in 2 dimensions I've come to the following integral, from which I can extract the proper ultraviolet behavior ($a \ll 1$) of the theory: $\int_0^\infty e^{-(4+a^2)x}\left[...
Misora Grilo's user avatar
4 votes
1 answer
169 views

Problems differentiating four-vectors to find the equation of motion using Euler-Lagrange equations.

This post directly follows this question and is very similar in nature: Consider the following Lagrangian: $$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\...
Sirius Black's user avatar
3 votes
0 answers
110 views

Integrate $ I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg) $

I am currently working on a mathematical problem in QFT when I came across the integral: $$ I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg) $$ I have no idea how to do such an integral, or where to ...
Gradient137's user avatar
2 votes
1 answer
330 views

Question on complexification of $\mathfrak sl(2,\Bbb C)$

As six generators of the real Lie algebra $\mathfrak sl(2,\Bbb C)_\Bbb R$ I can use the Pauli matrixes as follow: $X_1=\frac{1}{2} \sigma_1, X_2=\frac{1}{2} \sigma_2, X_3=\frac{1}{2} \sigma_3$ $Y_1=\...
Andrea's user avatar
  • 147
2 votes
3 answers
194 views

How to give a closed form to $e^{a(x) \frac{d}{dx} + b(x)I}[f]$ in physicists style abuse of notation?

In Quantum Mechanics we have the famous time evolution result (here $a$ is a constant) $$ e^{a \frac{d}{dx}}[f] = f(x+a) $$ Which is an abuse of notation but makes sense due to Taylor's Theorem. In ...
Sidharth Ghoshal's user avatar
2 votes
1 answer
74 views

Using Euler-Lagrange equations show that the following EoM can be written as $D_\mu D^\mu\phi+m^2\phi + \lambda\left(\phi^\ast\phi\right)\phi=0$

This post is a follow-up to this previous question. Consider the following Lagrangian: $$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\nu-\partial^\nu A^\...
Sirius Black's user avatar
1 vote
0 answers
51 views

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$...
Tuneer's user avatar
  • 161
1 vote
1 answer
85 views

How can I plot this function?

I found this function while looking at quantum field theory. Defined by: $$K(x,y) = \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} e^{i(x k + y q)}\sqrt{ (m^2 + k^2 + q^2) } dk dq$$ ...
zooby's user avatar
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1 vote
0 answers
249 views

Formula for calculating symmetry factors for Feynman diagrams?

In a homework exercise for a QFT course I'm taking, we have been tasked with calculating the integral $$ Z(g)=\int_{-\infty}^{+\infty} dx\, e^{-\frac{1}{2}x^2 + \frac{g}{3!}x^3} $$ up to order $g^4$ ...
HelloGoodbye's user avatar
1 vote
1 answer
342 views

Asymptotic expansion about branch point

If we have an analytic function but which has a branch point (yes technically it's only analytic in open sets which are disjoint with a branch cut but that's besides the point) is there a way to ...
thedoctar's user avatar
  • 133
0 votes
1 answer
512 views

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 ...
B. T.'s user avatar
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