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Results tagged with quantum-field-theory
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user 100922
Quantum Field Theory (QFT) is the theoretical framework describing the quantisation of classical fields which allows a Lorentz-invariant formulation of quantum mechanics. QFT is used both in high energy physics as well as condensed matter physics and closely related to statistical field theory. Use this tag for many-body quantum-mechanical problems and the theory of particle physics. Don’t combine with the [quantum-mechanics] tag.
1
vote
Can a scalar field transform nontrivially under a local special conformal transformation?
I don't completely understand the question. How a scalar transforms is completely dictated by conformal symmetry. The transformation law is
$$K_\mu \phi(x) = \big(\Delta x_\mu + x_\mu \, x_\nu \parti …
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1
vote
Spatial position and 3 momentum relation between particle states in canonical quantization (...
The answer is really a computation. It doesn't make sense to talk about the position of a state; there's no operator $X^\mu$ to probe it. At best you can insert local operators and see what happens. I …
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1
vote
Accepted
Interacting term in Jellium model
Suppose that you have two quartic terms with different orderings, like $a^\dagger a^\dagger a a$ and $a^\dagger a a^\dagger a $, assuming that total momenta and Lorentz spins etc. of the oscillators …
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2
votes
Existence of lagrangians at strong coupling
Free theories can be built out of non-interacting scalars, fermions and vectors, and therefore have a Lagrangian description. There may be exceptions for higher-spin fields or exotic SUSY multiplets e …
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1
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What's the relation between Euclidean and Minkowski entities in lattice field theory?
Real $\tau$ does not reflect all of the physics of Minkowski spacetime. But there are many physical observables that do not depend on the choice of signature. For instance, if you measure a two-point …
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2
votes
Dimensional regularisation in $\phi^4$ theory
This is just bookkeeping. The correction to $\lambda$ is
$$\delta \lambda \sim \lambda^2 \times \text{loop integral} = g^2 (\mu^{4-d})^2 \times \text{loop integral}$$
so
$$\delta g \sim g^2 \mu^{4-d} …
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1
vote
Accepted
Why does the divergence of a QFT's coupling constant under RG flow trivialize the theory if ...
The description of "trivialization" is outdated and not very sharp. In a more modern language, it just means that the low-energy description you have, say a $\phi^4$ theory in $4d$, is not by itself e …
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2
votes
Accepted
Generators of conformal transformations change of basis
The Greek indices $\mu,\nu,\ldots$ are the usual spacetime indices, so there are $d$ of them. In the conventions of your book, they run from $1,2,\ldots,d$. Likewise $\eta_{\mu \nu}$ is a $d \times d$ …
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1
vote
Accepted
Counterterms cancelling divergences
You say that "All the $M$'s in the two-point function will be replaced by $M_\text{ren}$" but that's too fast. By definition, the renormalized mass is something like
$$
M_\text{ren}^2 \equiv \lim_{p^2 …
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0
votes
Why do we demand that the counterterms in $\varphi^3$ theory be $O(g^2)$?
There's a cute argument based on symmetry, at least at when $Y = 0$ is treated as a perturbation. The $g=Y=0$ theory has a $\mathbb{Z}_2$ symmetry $\phi \to -\phi$. Let $X$ be some observable that's n …
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1
vote
What is free energy in the context of a quantum field theory?
The theories on question are local QFTs in 3d. You can put such a theory on any compact 3-manifold $M$, and then you can compute observables such as the partition function $Z[M]$ or correlation functi …
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13
votes
Accepted
Can you naïvely reduce the dimensionality of a QFT?
Yes, you could consider such a dimensional reduction. The idea is to decompose $\phi(t, \mathbf{x})$ as
$$
\phi(t, r, \theta) = \sum_{k \in \mathbb{Z}} \phi_k(t, r) e^{i k \theta}.
$$
You plug this An …
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