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.

Quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics and is used to construct physical models of subatomic particles in particle physics/high-energy physics, quasiparticles in condensed matter physics and highly relevant to statistical field theory.

QFT treats particles as excited states (also called quanta) of their underlying fields, which are — in a sense—more fundamental than the basic particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding fields. Each interaction can be visually represented by Feynman diagrams, which are formal computational tools, in the process of relativistic perturbation theory.

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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 $\phi : \mathcal{S}(M)\to \mathcal{L}(\mathcal{H})$…
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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 physics. Unlike many other fields in pure…
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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 Schroedinger equation: $$i\hbar\dfrac{\partial}{\partial…
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In the Physicists' definition of the path integral, does the result depend on the choice of partitions?

The standard definition of the path integral in Quantum Mechanics usually goes as follows: Let $[a,b]$ be one interval. Let $(P_n)$ be the sequence of partitions of $[a,b]$ given by $$P_n=\{t_0,\dots,t_n\}$$ with $t_k = t_0 + k\epsilon$ where…
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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 applications of TQFTs within both disciplines…
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Renormalization for mathematicians

Can someone explain to me the processes of renormalization and regularization used in quantum field theory and similar fields in a way that a pure mathematician might make sense of it? Is there a mathematically rigorous way to go about doing them?…
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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 operator $\cal{A}?$ How mathematically rigorous is the…
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Mathematics and Physics prerequisites for mirror symmetry

I am a physics undergrad interested in Mathematical Physics. I am more interested in the mathematical side of things, and interested to solve problems in mathematics inspired by physics maybe with the help of techniques in Physics. My current…
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Are GNS representations the way to build physical Hilbert spaces?

Consider a separable $C^*$ algebra $\mathcal A$. The space of states is also separable in the weak* topology, let $S$ be a countable dense subset. Denoting with $H_\omega$ the GNS representation of a state $\omega$ we retrieve a representation of…
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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 functor from the Segal category $\mathcal{C}$ to…
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What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ and am especially interested in its relation to…
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Reference for rigorous treatment of the representation theory of the Lorentz group

When I studied representation theory for the first time it was only focused on finite groups. It was the second half of a one semester course in group theory, and the book employed was "Representation Theory: A First Course" by Harris and…
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Path (Feynman) Integrals over Graphs

I was thinking about Feynman integrals the other day and in particular about discretizing the paths. Does anyone know the lay of the land about what happens when you do path integrals over, say, a lattice or graph?
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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: “the square root cut starting at $±im$ tells us that…
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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},\ldots,x_{n}\right)\, {\rm d}x_{1}\,{\rm…
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