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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.
17
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Cutoff-Scheme Renormalization and Order of Integration in QFT
The following is the result of Fubini's Theorem, describing when you can replace a double integral with an iterated integral safely:
For a set $X \times Y \subset \mathbb{R}^2$, if $\iint |f(x,y)| d( …
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9
votes
2
answers
2k
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Propagator Correction in $\phi^4$ theory - why doesn't this secular growth break perturbatio...
The free propagator for a massive $m\neq0$ real scalar field is the following:
$$
G_{0}(x,y) \ = \ \int \frac{d^{4}p}{(2\pi)^4} \frac{e^{i p \cdot (x-y)}}{p^2 +m^2 - i \epsilon}
$$
It is a well-know …
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9
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1
answer
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Matsubara Field Theory - what does imaginary time $\tau$ in $G(\tau,\mathbf{x})$ mean?
Consider the free, real scalar field $\phi$ in Matsubara Finite-Temperature quantum field theory, where our system is kept in equilibrium with a heat bath at temperature $\frac{1}{\beta}$.
Then the f …
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7
votes
1
answer
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Feynman Diagrams for $\phi^{4}$ theory, up to order $g^2$
I'm considering $\phi^4$ theory with the action $S[\phi] = S_{\mathrm{FREE}}[\phi] + \frac{g}{4!} \int d^{4}x \ \phi(x)^4$.
I'm supposed to come up with the Feynman diagrams up to order $g^2$ for the …
7
votes
1
answer
417
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Relation between the Hadamard Expansion and the vacuum state in question?
I am trying to understand the paper "Off-diagonal coefficients of the DeWitt-Schwinger and Hadamard representations of the Feynman propagator" by Decanini and Folacci.
Consider a ($m$)assive scalar fi …
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7
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1
answer
1k
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How do you write the Wightman function $\langle\phi(t_1)\phi(t_2)\rangle$ for a massive scal...
For a free real scalar field $\phi(t,\mathbf{x})$, we define the Wightman function as:
$$
W(t_1,t_2) \equiv \langle 0 | \phi(t_1,\mathbf{x}_1) \phi(t_2,\mathbf{x}_2) | 0 \rangle
$$
I'm suppressing the …
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7
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2
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$\phi^{4}$ Propagator - Feynman Diagram: internal vertex that loops back to itself
In all that follows I'll be dealing with everything massless.
The free, massless propagator ($\mathcal{L} = \int d^{4}x \left(\partial \phi(x) \right)^{2} $) is supposedly given by $G_{0}(x,y) = c (x …
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6
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1
answer
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Klein-Gordon Inner product being independent of the choice of spacelike hypersurface $\Sigma...
Here I'll work in flat 4-dimensional Minkwoski space, but using arbitrary coordinates (described by some metric $g_{\mu\nu}$).
Suppose we've got two complex-valued scalar functions $f$ and $g$ which …
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6
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1
answer
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Does the Fock space for a free QFT decompose into a tensor product of Fock spaces for each m...
Take a free relativistic QFT, say for a real scalar field $\phi$ with the Lagrangian density
$$
\mathscr{L} = \frac{ \partial_{\mu} \phi \partial^\mu \phi - m^2 \phi^2}{2} \ .
$$
After quantization we …
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6
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2
answers
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double line notation (three and four gluon vertices) - how is this a simplification?
This is closely related to my previous post Double line notation - gluon propagator
I'm trying to understand the double-line vertices for the gluon in the case of a $U(N)$ gauge group. Normally, the …
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6
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1
answer
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For Hawking radiation, is the scalar field $\phi$ assumed to be in the Unruh vacuum state?
In Hawking's paper "Particle Creation by Black Holes" I'm not really able to pick apart what vacuum state Hawking is assuming the field $\phi$ to be in.
The paper "Hawking radiation as perceived by di …
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5
votes
1
answer
538
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Wick-rotating the Fourier transform of $\mu+1$ propagators
In Equation (8) of this paper by Groote et. al., we are given the following Euclidean identity:
$$
\int \frac{d^{4}\mathbf{p}_{\mathrm{E}}}{(2\pi)^{4}} \frac{e^{ i \mathbf{p}_{\mathrm{E}} \cdot \mathb …
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5
votes
0
answers
376
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What motivates the definition of the Klein-Gordon Inner Product? [duplicate]
I am following along Marc Casal's lecture slides "Quantum Field Theory in Curved Spacetime". For scalar functions $f$ and $g$ we define the Klein-Gordon inner product as follows:
$$
\langle f,g \rangl …
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5
votes
1
answer
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Can you expand a real scalar field $\phi(t,\mathbf{x})$ in terms of spherical harmonics?
A massless real scalar admits the expansion
$$
\phi(t,\mathbf{x}) = \int \frac{d^3\mathbf{p}}{(2\pi)^{3/2} \sqrt{2|\mathbf{p}|}} \bigg( e^{ - i |\mathbf{p}| t + i \mathbf{p} \cdot \mathbf{x} } a_{\mat …
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5
votes
1
answer
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How does the normalization of an $n$-particle state $|n_{\mathbf{k}}\rangle$ work?
You can expand the free, real scalar field in the following manner
$$
\phi(x) = \int \frac{d^{3}\mathbf{k}}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{k}}}} \bigg[ e^{- i \omega_{\mathbf{k}}x^0+i \math …
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