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Results tagged with quantum-field-theory
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user 71434
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.
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Creation and Annihilation operators of negative momenta in QFT
I'm studying the free Klein-Gordon field, $\phi$. You can write this field as:
$\phi(\mathbf{x}) = \int \frac{ d^{3} \mathbf{p} }{(2\pi)^{3}} \frac{1}{\sqrt{ 2 E_{\mathbf{p}} }} \left[ a_{\mathbf{p}} …
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What motivates the definition of the Klein-Gordon Inner Product? [duplicate]
I am following along Marc Casal's lecture slides "Quantum Field Theory in Curved Spacetime". For scalar functions $f$ and $g$ we define the Klein-Gordon inner product as follows:
$$
\langle f,g \rangl …
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Are correlation functions $\langle \phi(x_{1}) \cdots\phi(x_{n}) \rangle $ invariant under c...
If I have a correlation function $$\langle \phi(x_{1}) \phi(x_{2}) \cdots \phi(x_{n-1}) \phi(x_{n})\rangle ,$$ can I cycle through the fields?
I'm not sure I'm using the correct terminology. But for …
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Perturbing about the Renormalized Field Theory: What justifies perturbing if the counter-ter... [duplicate]
I have a very basic question about Renormalization in Quantum Field Theory. Consider the following passage (about $\phi^4$ theory) from Zee's Quantum Field Theory in a Nutshell (from Chapter III.3):
…
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For real scalars $\phi$ and $\partial_t \phi = \pi$, why can't the Hamiltonian have terms of...
Suppose one has a real scalar field $\phi$ and its conjugate momentum field $\pi := \partial_t \phi$. If the scalar is free and using metric $(-+++)$, the action is
$$
\int d^4x\ \mathscr{L}_{\mathrm{ …
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2
answers
358
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Real scalar field and its quantum state: why are the diagonal components static here?
Consider a (free and massless) real scalar field $\phi(x)$ with Hamiltonian
$$
H := \int d^3\mathbf{x}\; \bigg[ \frac{1}{2} \pi^2(\mathbf{x}) + |\nabla\phi(\mathbf{x})|^2 \bigg]
$$
where $\pi(\mathbf …
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1
answer
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Wightman Function for complex scalar field - timelike separations?
For a complex scalar field $\Phi$, the field has the expansion
$$
\Phi(x^0,\mathbf{x}) = \int \frac{d^{3}\mathbf{p}}{\sqrt{ 2 E_{\mathbf{p}} (2\pi)^3 } }\ \bigg[ e^{- i E_{\mathbf{p}}x^0 + i \mathbf{p …
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is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x...
Is the following statement true?
$$
\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)
$$
where $\mathscr{P}$ is the Cauchy principal value. The above i …
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What effect does multiplying $\mathscr{L}$ by $-1$ have on the propagator?
I am following along Ashok Das' development of Thermofield dynamics in his book Finite Temperature Field Theory. Here you have two real scalar fields $\phi_1$ and $\phi_2$ with Lagrangian density
$$
\ …
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1
answer
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Why do Candelas and Howard say that $\sum_{n=1}^\infty \cos\left( n \kappa \epsilon \right) ...
In the paper Vacuum $\langle \phi^2 \rangle$ in Schwarzschild Spacetime by Candalas and Howard, they say that for each non-zero $\epsilon$ it is true that
$$
\sum_{n=1}^\infty \cos\left( n \kappa \eps …
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Accepted
Coupled Oscillator's Stiffness and speed of light
First I re-write your equation as
$$
\partial_{0}^2 \phi(x) - v^2 \sum_{j=1}^3 \partial_{j}^2 \phi(x) + \Omega_0^2 \phi(x) = 0
$$
Plugging in $v=c$ and $\Omega_0 = mc$ your equation is
$$
\partial_{0 …
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Accepted
Klein-Gordon Propagator for spatial separation $x - y = (0, r)$
The easiest way to proceed is to notice that the first integral can only depend on $r := |\vec{r}|$. To convince yourself of this, you can rotate $\vec{r}$ by any rotation $R \in SO(3)$, and you notic …
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Schrodinger versus Interaction Pictures for Mukhanov Field $\hat{v}(\eta,\mathbf{x})$ in inf...
EDIT: This is more than anything else, a question about how to define Schrodinger-picture operators, if you are given an Interaction picture set of operators (with a time-dependent potential: in this …
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0
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Out-Of-Time-Correlators for free and interacting field theories
Consider the Out-Of-Time-Correlator (OTOC)
$$
\mathrm{Tr}\left[ - [ \phi(t,\mathbf{x}) , \phi(t',\mathbf{x}') ]^2 \rho\ \right]
$$
where $\rho$ is some initial density matrix, and $\phi$ is a quantum …
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2
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0
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Normalization for the overlap $\langle \phi_a | 0 \rangle$
This question is related to my previous post here.
According to Wienberg's Volume I (9.2.9), we have the result
$$
\langle \phi_a | 0 \rangle = \mathcal{N} \mathrm{exp}\left( - \frac{1}{2} \int d^{3} …
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