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Results tagged with quantum-field-theory
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user 232440
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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Vanishing terms in calculating a simple propagator in QFT
In David Tong's lecture notes on quantum field theory, at the top of page 38, we calculate the amplitude for a particle to propagate from $y$ to $x$:
$$\begin{align}\langle0|\phi(x)\phi(y)|0\rangle&=\ …
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In what way are the Hilbert spaces of non-interacting quantum field theories different from ...
In QFT we use, rather than a single Hilbert space, $\mathcal H$, a Fock space$^1$: $$F_v(\mathcal H)=\bigoplus_{n=0}^\infty S_v\mathcal H^{\otimes n}, \tag{1}$$ which allows for states to exist with a …
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What is the meaning of a cross section having "no angular dependence"?
This is in reference to page 112, equation 4.99 of Peskin and Schroeder.
We have just completed the first calculation (to first order) of a differential cross section. Specifically that of 2-particle …
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What prevents two quantum fields from coupling?
In David Tong's lecture notes on quantum field theory, at the bottom of page 138 he states that, regarding the coupling of the electromagnetic field to a real scalar field, there exists "no suitable c …
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On the creation of wave packets with particular properties in quantum field theory
At the start of chapter 5 of Mark Srednicki's lecture notes on quantum field theory we define an operator that creates a particle that is "localised in momentum space near $\mathbf {k_1}$, and localis …
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Why implementing $\partial_\mu A^\mu=0$ as an operator equation is not valid?
This questions relates to equation (4.43) in Timo Weigand's QFT lecture note (page 108). Weigand makes the claim that $\partial_\mu A^\mu=0$, interpreted as an operator equation, causes issues due to …
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Delta function squared in Weigand's QFT notes
On page 74 of Timo Weigand's QFT notes, right at the top, the following equality is used:
$$\left[(2\pi)^4\delta^{(4)}(p_f-p_i)\right]^2=(2\pi)^4\delta^{(4)}(p_f-p_i)(2\pi)^4\delta^{(4)}(0) \tag{2.167 …
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The quantisation of the harmonic oscillator applied to the free Klein-Gordon field
In David Tong's lecture notes on quantum field theory, at the bottom of page 23, we are applying the quantisation of the harmonic oscillator to the field to obtain expressions for the field operators …
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Ordering ambiguity in the Feynman propagators obtained using Wick's theorem
Applying Wick's theorem to a string of four field operators, $\phi_a\equiv\phi(x_a)$:
$$T(\phi_1\phi_2\phi_3\phi_4)=\{...\}, \tag{1}$$
we obtain several terms, three of which are fully contracted fiel …
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Why is there a negative sign in the time evolution operator when defining in/out states? (Pe...
This relates to Peskin & Schroeder's QFT book, equation 4.70 on page 104.
To define in and out states we take our initial state and evolve it far into the past, and do the same for our final state. Pe …
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Are particles literally waves or just abstract probability waves?
It's tempting to try to interpret the mathematics of QM or QFT too literally, however this can be a dangerous game to play. Quite often you will cause more problems for yourself than you will solve by …
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Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder)
This relates to the bottom of page 83 in Peskin and Schroeder.
The following claim is made:
At any fixed time $t_0$ we can of course expand $\phi$ in terms of ladder operators
$$\phi(\textbf{x},t_0)= …
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Question on how to actually use momentum space Feynman rules in $\phi^4$-theory
The momentum space Feynman rules state that we "integrate over all undetermined momenta" and "impose momentum conservation at each vertex". This is given for example on page 95 of Peskin and Schroeder …
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Correlation Function and quadrivectors, how to simplify a triple integral?
Instead of doing an integral of $d^3p$ over all of $\Bbb R^3$ you can switch to spherical polar coordinates and integrate the sphere over all of $r$ (which in this case they've just called $p$ again). …
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If the time-ordering symbol does not respect commutation relations, why is the expansion of ...
A standard result in QFT is the expression of the interacting theory correlation functions in terms of field operators in the interaction picture and the free theory vacuum:
$$\langle\Omega|\mathcal T …
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