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.
207 questions with no upvoted or accepted answers
54
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0
answers
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How are topological invariants obtained from TQFTs used in practice?
Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places:
Atiyah, Topological quantum field theory
Lurie, Topological Quantum Field Theory ...
- 661
26
votes
0
answers
854
views
Was von Neumann's 1954 ICM address "Unsolved Problems in Mathematics" outdated?
I recently tried to "explain" the generalized probability theory aspect of quantum theory (as one common part of both quantum field theory and quantum mechanics), in the sense of motivations for the ...
- 7,536
14
votes
0
answers
485
views
What are D-branes (in a topological field theory)?
In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
- 351
10
votes
0
answers
379
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},\...
- 487
9
votes
0
answers
251
views
Motivation for quantum cohomology rings
I can't seem to find a good source for the motivation for defining the big quantum cohomology ring with its quantum product. Collecting the Gromov-Witten invariants in a generating function seems like ...
9
votes
0
answers
481
views
What are the mathematical foundations of the renormalisation group?
Briefly, RG refers to mathematical tools that allows systematic investigation of the changes of a physical system as viewed at different distance scales. These methods are very important in quantum ...
user93511
9
votes
1
answer
790
views
In what sense is quantum field theory mathematically incomplete?
Is the Yang-Mills existence and mass gap (Millenium Prize problem) essentially what is required?
Or are there more problems in putting QFT on strong mathematical foundations?
For example, the ...
- 91
7
votes
0
answers
190
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 ...
- 243
7
votes
0
answers
279
views
Reference for 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 ...
- 30.3k
6
votes
0
answers
627
views
Can anyone help to perform this heinous integral? (Peskin & Schroeder's Quantum field theory (4.76)-(4.78)) (Including my own trial )
I am reading the Peskin & Schroeder's An introduction to Quantum field theory, p.105~p.106 (Construction of a cross-section from the invariant matrix element (p.104) ) and stuck at understanding ...
- 3,186
6
votes
0
answers
512
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Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?
Edition
On July 15 2021, the description of the question has been considerably modified to meet the requirement of making this question more OP-independent and thus more useful for general readers. ...
- 203
6
votes
0
answers
457
views
Is there a relation between Super Riemannian manifolds and Kähler manifolds?
(This question has a physics motivation).
Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
- 772
5
votes
0
answers
211
views
Are creation and annihilation operators special?
In Weinberg's The Quantum Theory of Fields,volume I, the author quotes a theorem that left me a bit mystified. He states
Any operator $O: \mathscr{H} \rightarrow \mathscr{H}$ may be written
$$O=\sum_{...
- 1,749
5
votes
0
answers
1k
views
Geometric algebra and quantum field theory
What 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 ...
- 941
4
votes
0
answers
153
views
How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines?
Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$.
Is it possible to do so ...
- 2,763
4
votes
0
answers
415
views
In what situations are Path Integrals well-defined?
In physics I have come across contexts where apparently path integrals are well-defined, and others where they are not. However I have no clear understanding of when and why they succeed or fail to be ...
- 351
4
votes
0
answers
142
views
Calculation involving hypergeometric functions
I am trying to derivethe $N$ defined via the following integral:
$$\displaystyle \frac{4\pi^2}{\omega N^2}= \int_0^L \frac{dz}{z}
\left(1-\frac{z^2}{L^2}\right)^{-1} |V_k(z)|^2 $$
where
$$\...
- 127
4
votes
0
answers
99
views
What is $H_{3}Spin(3)$, and how is this related with the twist of framing on a 3-manifold?
From the question, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action
$$I(g)=\frac{1}{4\pi}\int_{M}\mathrm{Tr}(\omega\wedge d\omega+\frac{2}{3}...
- 517
4
votes
0
answers
234
views
4-dimensional integral representation of modified Bessel function
In 4-dimensional Euclidean QFT, the propagator can be expressed in terms of a modified Bessel function of the second kind:
\begin{align}
\Delta_E (x_1-x_2)&\equiv\int\frac{d^4p}{(2\pi)^4}\frac{e^{...
- 427
4
votes
0
answers
301
views
Instanton Moduli Space on ALE Spaces?
I've recently been learning about moduli spaces of instantons on $\mathbb{C}^{2}=\mathbb{R}^{4}$. From what I can gather, one can consider the framed moduli space of torsion-free sheaves on $\mathbb{...
- 2,614
4
votes
0
answers
102
views
Non-negative integral implies non-negative integrand (up to total derivatives)?
Take a functional $S[\varphi]$ on a $d$-dimensional space-time of the form
$S[\varphi]=\int d^d x\, L(x,\varphi,\partial_\mu \varphi,\partial_\mu \partial_\nu \varphi, \dots)$
where $\varphi(x)$ is ...
- 41
4
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0
answers
1k
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Is the physicist's notion of the generalized Dynkin index of a representation of a Lie-Algebra well defined?
Let there be a Lie-Algebra $\mathfrak{g}$ (I am most interested in the cases, where $\mathfrak{g}$ is simple or semisimple) and a irreducible representation R. Let $t^a$ be a base of the Lie-Algebra ...
- 523
4
votes
0
answers
65
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 statement-...
- 3,875
4
votes
0
answers
638
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 ...
Fan
4
votes
0
answers
1k
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 \frac{ke^{ik\...
- 1,097
4
votes
0
answers
295
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 ...
- 41
4
votes
0
answers
154
views
4D TQFT construction from a modular tensor category
I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category.
I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
user27329
4
votes
0
answers
271
views
Steps toward making the Feynman integral rigorous
I would like to know if there has been any progress recently toward giving the Feynman integral a rigorous definition. I heard that Richard Borcherds is working on giving quantum field theory a ...
- 296
3
votes
0
answers
105
views
Properties of reducible representations
I have the following doubt. Let's assume we have two mixed states $\rho_1 = \Sigma_i a_i \omega_i^{1}$ and $\rho_2 = \Sigma_i b_i \omega_i^{2}$ on the same algebra, where the states $\omega$ are all ...
- 199
3
votes
0
answers
201
views
Propagators and PDEs
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. Mathematically, these derivations are somewhat magical (typically one ...
- 39
3
votes
0
answers
85
views
Derivative after applying residues theorem, that removes original poles
I have a question related to a typical integration in particle physics.
Suppose one has a function $f(t) = \dfrac{i}{t^2-\omega^2 }$ with $t, \omega \in \mathbb{R}$ and $\omega>0$. I am interested ...
- 31
3
votes
0
answers
110
views
Integrals of functions with poles on the real axis (Sokhotski–Plemelj theorem) in the sense of Distributions/Generalized Functions.
The Sokhotski–Plemelj theorem [Wikipedia link] states that
$$
\lim_{\epsilon \to 0 } \int_{\mathbb R} dz \; \frac{f(z)}{z \pm i \epsilon} = \mp i \pi f(0) + \mathcal{P} \int dz \; \frac{f(z)}{z} \;,
...
- 61
3
votes
0
answers
74
views
How to solve this integral for three-point correlation function?
I need to solve the following integral as part of the computation of a three-point correlation function in a charged massless quantum field theory, where I can also assume conformal invariance.
$$
\...
3
votes
0
answers
77
views
Why does it seem possible from Physics to say something about orientability locally?
In Quantum Field Theory, if I understand correctly, Physicists showed that certain process are not symmetric under time reversal. This should be a local thing, mathematically, and would show that ...
- 291
3
votes
0
answers
1k
views
Prerequisites for Quantum Fields and Strings: A Course for Mathematicians
I'm interested in learning more about the mathematical structure underneath of quantum field theory and string theory. I've taken a few courses on quantum field theory before, so am getting more ...
- 351
3
votes
0
answers
358
views
n-dimensional Gaussian integral with absolute values
Given two matrix $A$ and $D$ and a column vector $x$, what is the value of the following integral?
$\int d^nx \; \; e^{x^T A x + \mid x \mid^T D \mid x \mid + B x}$
where $\mid x \mid_i = \mid x_i \...
3
votes
0
answers
101
views
Quantum Groups for Generic q and 3d-TQFT. What breaks?
I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the ...
- 121
3
votes
0
answers
117
views
Integrate $ I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg) $
I am currently working on a mathematical problem in QFT when I came across the integral:
$$
I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg)
$$
I have no idea how to do such an integral, or where to ...
- 163
3
votes
0
answers
272
views
What is the difference between a massless pinor and a spinor?
I was reading the paper "The Pin Groups in Physics: C, P, and T" by M. Berg, C. Morette-DeWitt et al. in which they analyze the (double) covering groups of (Lorentzian) orthogonal groups $\...
- 1,815
3
votes
0
answers
496
views
Explicitly proving that the Hamiltonian is Lorentz covariant
I want to show explicitly that the Hamiltonian
$$
H = -\Omega V + \int d\tilde{\textbf{k}}\ \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) )
$$
is Lorentz ...
- 2,794
3
votes
0
answers
88
views
What is the relation between BRST quantization and gauge fixing quantization
To quantize gauge field, one usually use gauge-fixing procedure and then plus ghost field, my question is what the relation between BRST quantization and gauge fixing quantization is? Because the ...
- 1,211
3
votes
0
answers
99
views
Is it possible to give one general definition of the normal ordering symbol?
In Quantum Field Theory one usually defines the Normal Ordering Symbol by means of examples and a description of its action: the normal ordering $N$ applied to one expression will be the expression ...
- 27.4k
3
votes
0
answers
110
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:
(...
- 2,243
3
votes
0
answers
91
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 ...
- 131
3
votes
0
answers
98
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 ...
- 896
3
votes
1
answer
321
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 ...
- 51
3
votes
0
answers
144
views
Wilson lines, boundary condisions, surface defects of TQFTs
I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
user27329
3
votes
0
answers
291
views
Could Motives aid in the study of the Navier-Stokes equations?
Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these ...
- 7,456
2
votes
0
answers
146
views
Separability of functional space
Perhaps the discussion of my question is too specific. So, let me make it more general.
Consider a set of functions $q:\mathbb{R}^3\rightarrow\mathbb{R}$ represented as $q(\mathbf{k})$ where $\mathbf{...
- 184
2
votes
1
answer
104
views
How should we think about functional differentiation
I've been introduced to the following notion of functional differentiation:
Let $F: \Phi \rightarrow \mathbb{R}$ be a linear functional on $\Phi$ (understood as a space of sections over a manifold $M$...
- 1,749