New answers tagged quantum-field-theory
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Insertions of the action inside correlators
I'll do this in Euclidean space. The partition function is
$$
Z[\alpha] = \int e^{-\alpha S}
$$
Here $\alpha$ is some overall factor; could be $1/\hbar$, or inverse temperature $\beta$ depending on ...
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2
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Why does the imaginary time Euler-Lagrange equation imply the potential goes to zero at infinite imaginary time?
For the shape of the potential $V$ Ref. 1 is apparently referring to Fig. 2, i.e. the potential $V$ is assumed to have classical false vacuum, i.e. a local minimum $V(q_0)=0$ that is metastable, i.e. ...
- 207k
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$\phi^4$ quantum fields theory with vanishing physical mass
Assume spacetime dimension $2 \leq d \leq 4$. The $\phi^4$ coupling constant $\lambda$ has mass dimension $[\lambda] = 4-d$. The one-loop mass correction is linear in $\lambda$. Given the UV cutoff ...
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Non-crossing approximation (NCA) in 'Large-$N$ expansion' (Altland & Simons CMFT)
Let me just write down a proper answer. I will address the questions in reverse order.
The wavy line is a notational device that splits the four-point vertex into two sides. Given the interaction
$$
...
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Wave packet as a field configuration acting like a particle's wave function?
There is a mathematical equivalence between two different physical situations.
The first situation is a wavepacket in a classical field. This describes some localized packet of energy in an ordinary ...
- 51.2k
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Where does the energy in a fundamental interaction come from?
In your electron example, there are two problems:
Although independently both electrons gain energy, you would never find two electrons close to one another to begin with, because such repulsion ...
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Wave packet as a field configuration acting like a particle's wave function?
Before the sentence you quoted, Caroll writes:
Think of a state in which just one mode is involved, the mode with $k = 0$. Since $k = 2\pi/\lambda$, that’s a mode with “infinite wavelength”—basically ...
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Confusion about solution of the Callan-Symanzik Equation
I think it is just because we integrate backward in time, $t'$ is the time backward, and the "initial" concentration is $D(t, x)$, and the final concentration is $D_i(x)$ , and as time goes ...
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Beta function calculation in massless minimal subtraction $\phi^4$ theory
First: you have an incorrect sign in your $\delta_\lambda$: see, for example, David Skinner's AQFT notes. In your notation, you should have that the quartic counterterm is positive
$$
``\delta_\lambda ...
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Unitarily Inequivalent Representations
For a succinct explanation of the orthogonality of the vaccua for the van Hove model (which seems to be what the initial pdf might have been referring to), see Section 2 of this paper.
Tne answer by ...
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QFT with massless particles
When textbooks specifically mention massless particles, they always mean renormalized mass. Unless protected by some type of symmetry, if the bare mass vanishes, the renormalized mass does not (due to ...
- 26.6k
10
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Photon propagator in path integral vs. operator formalism
Right off the bat, to answer your first question $\Pi^{\mu\nu}(p)$ is the Fourier transform of $\langle 0| T\{A_{\mu}(x) A_{\nu}(x') \} |0 \rangle$. Usually $\Pi^{\mu\nu}(p)$ would be referred to as ...
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Intuitive reason why bound states correspond to poles
I think this link will help you:
https://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Scattering_III.htm
... if the scattering matrix $S_0(k)$ becomes infinite at some
complex value of $k$, ...
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How to avoid the ordinary Coulomb solution in QCD?
First, I think your proposed current would break gauge invariance, but that's a relatively trivial problem in that I think you could reformulate your question getting around that issue.
The bigger ...
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3
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$ \pi^0\to \gamma\gamma$ parity conservation
As @Cosmas Zachos said in his comment, the Levi-Civita pseudotensor $\epsilon$ has negative parity. But why?
This is the case because $\epsilon_{\mu\nu\rho\sigma} := \sqrt{-g}\ \varepsilon_{\mu\nu\rho\...
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Missing counterterms in $\phi^3$ + $\phi^4$ theory in 1PI effective action
First of all, this 1PI thing is quite comparable to perturbation series. Since you use Peskin/Schroeder, check out the reference to that Coleman Weinberg potential project: Coleman and Weinberg in ...
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Loop Effect of $\phi$ Propagator in $t$-channel of scalar $\phi^3$ theory
You can choose a branch cut to evaluate the negative log if I remember correctly e.g. $\log(-x) = \log(|x|) + i \pi$. The imaginary part is related to the virtual particles going on shell. This is a ...
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Can the Strong CP problem be explained by the cancellation of quark electric dipoles by dipoles from some as yet undetected matter surrounding quarks?
The question is a bit confused about the relation between the strong CP problem and electric dipole moments (and so are some prior responses).
The strong CP problem concerns a certain field-...
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Can the Strong CP problem be explained by the cancellation of quark electric dipoles by dipoles from some as yet undetected matter surrounding quarks?
CP violation isn't about electric dipoles in quarks. It refers to an asymmetry in how particles and antiparticles decay, where one type might decay more often than the other. Electric dipole moments, ...
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Hermitian conjugation in Radial Quantization
I wish I could make this clear.
In the Euclidean QFT, we use the variable
\begin{equation}
z=e^{ix-\tau}
\end{equation}
to label the Eucludean quantum field $\phi_{E}(z)$, which is related to the ...
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Reference on partial wave expansion in the context of QFT
Apart from Weinberg QFT vol1, section 3.7, one can look at the paper: "On the general theory of collisions for particles with spin", Jacob and Wick 1959. A recent review that I found useful ...
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Phase Coherence in the BCS wavefunction and the Cooper Pair Wavefunction
First answer. This is a quantum theory: every amplitude is a priori a complex number. The most general expression would have all $u_{k}$ and $v_{k}$ complex. What you do to get that expression here ...
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Square of the Feynman amplitude for $a +b\to c+d$ and its reverse
This would only be true if the theory describing the scattering of ${a,b,c,d}$ is time reversal invariant.
It can be shown that if $S$ is the S-matrix operator, and $T$ is the time reversal operator, ...
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1
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Some calculation in Schwartz's Quantum field theory eq. (16.39)
What is definition of $\Pi_2^{\mu \nu}$? What do we call such an object?
It is defined in (16.24), the vacuum polarization tensor, after which the chapter is named.
What is $\gamma_E$?
As your ...
- 64.2k
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Where are quantised states in QFT?
Perhaps it is important to add one clarification to the already good answers here. Although the focus, as presented by the OP, is the difference between QM and QFT, it is important to point out that ...
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Where are quantised states in QFT?
Perhaps the most striking fact of quantum mechanics, and where the name comes from, is the fact that the energy of a quantum system is generally quantized (for bound states), i.e. it can only take ...
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Where are quantised states in QFT?
Particles are quanta, or discrete packets of energy, in a quantum field. The fact that we see photons (for example), with a minimum energy $E=hf$ for a given frequency $f$, and not a continuum of ...
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Understanding Feynman Diagrams in Loop Corrections to the propagator $\phi ^3 $ theory
...
And to be precise:
$$\tilde{\Delta }(k^2 )=\frac{1}{k^2 +m^2 -i\epsilon }\tag{14.3}$$
...
Why is (as Srednicki shows):
$$\int \frac{d^dl}{(2\pi )^d}\tilde{\Delta }((l+k)^2)\tilde{\Delta }(l^2 )$$
...
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Understanding Feynman Diagrams in Loop Corrections to the propagator $\phi ^3 $ theory
I was asked to expand my comment to an answer. Maybe it would be useful to keep in mind a more physical notion of what the loop diagrams represent.
Recall we are trying to understand our interacting ...
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Degenerate link variable configuration in $Z_2$ lattice gauge theory (Wen's QFT book)
The statement "the four-fold degenerate configuration are not globally gauge equivalent" means that these four configurations belong to four different equivalence classes.
To begin with, we ...
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What happens to the fermion spin when I move around it in a full circle
My confusion arised from the fact that the phase of the state gets a minus sign, not thr expectations value of the spin.
Trivial, but I must throw some of the responsibility on people that visualising ...
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How does inserting an operator in the path integral change the equation of motion?
The missing conceptual point to appreciate is that when you are calculating a correlation function it can be useful to interpret the same expression in different ways.
We wish to calculate the ...
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How does inserting an operator in the path integral change the equation of motion?
Yes, the Wilson line (2.42) is now part of the action in the path integral.
Yes, the symmetry defect operator (SDO) (2.41) is now part of the action in the path integral.
References:
T.D. Brennan &...
- 207k
3
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$2\pi$-rotation of fermionic states vs. fermionic operators
The point is that fermionic operators are not observables for many reasons. For instance they do not commute for causally separated arguments. You need two (an even number of) fermionic operators ...
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5
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Vacuum flucutuations = local entanglement between quantum fields?
By some measures the vacuum state is highly entangled as described by a result in quantum field theory called the Reeh-Schlieder theorem. See Section V of a review of "Quantum Information and ...
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2
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Path integral at large time
I'm assuming you mean immaginary time $T$. In this case the path integral is equal to the partition function (with $T=\beta$), as others have noted. Then, let the spectral decomposition of $H$ be
$$
...
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5
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Path integral at large time
Path integral and operator formalism correspondence states that for Euclidean time $t$, $$Z=\int D\phi e^{-S[\phi]}\leftrightarrow Z~=~{\rm Tr}_{\cal H}(e^{-t\hat{H}})$$
Fix an orthonormal basis $\{|n\...
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Path integral at large time
Briefly speaking, the standard argument is
A correspondence between the path integral formalism and the operator formalism in Minkowskian signature with unitary time evolution;
Wick rotate both ...
- 207k
2
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What does it mean to "resum" the large logarithms?
I don't even understand what do we mean with resumming
Resummation literally means re-summation.
The one-loop
$$
\Pi(p^{2})\propto log(p^{2}/\mu^{2})
$$
is the first non-constant term if we expand $\...
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vote
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2PI Effective Action from Double Legendre Transformation
It is difficult to find an explicit reference that really shows this property, but one can convince oneself from eq. (2.17) in PhysRevD.10.2428. Note here that the constant is the log of the free ...
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Classical Dirac equation
The distinction between quantum fields and classical trajectories arises from the different scales at which we observe and describe physical phenomena. In quantum field theory (QFT), both ...
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Derivation of two-body Coulomb interaction in momentum space
This is not possible to answer without knowing all of your source's relevant conventions, including the Fourier transform. One way you can check the book is look for a place where they use this ...
- 51.2k
1
vote
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2+1-dimensional $SU(N)$ Yang-Mills Theory
From the point of view of perturbation theory, QCD in lower dimensions is essentially identical to QCD in $d=3+1$. Namely, the beta function is negative, which means that the theory is UV complete, ...
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Checks of anomaly cancellation
A lot of the information you gave me is beyond my knowledge, not being a science professional myself or even a student. But I like reading physics textbooks from time to time and the chapter on ...
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Deriving the equal time anti-commutator of the Dirac fields
The relation you are trying to show is wrong. The equal-time anti-commutation relations for the Dirac field are (see for example Peskin & Schroeder, (3.108) )
\begin{align}
\big\lbrace\psi_a(\...
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Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?
Old post, but I just heard of the following analogy recently so I thought I'd share. In the comments and elsewhere on this site, people have given a good answer about the mathematical answer to why ...
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What is the energy of a photon in an electron-muon scattering?
Your reasoning is correct* and indeed
\begin{equation}
E_q=0 \,.
\end{equation}
Note however that only the external legs of the diagram correspond to the asymptotic states that can be measured, while ...
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Causality for gauge dependent operators in quantum field theories
Correct. As an example of how causality cannot be imposed on the electromagnetic potentials, consider the Coulomb gauge. In this gauge the propagation is instantaneous.
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Checks of anomaly cancellation
... if 𝐺 is a global symmetry of the classical Lagrangian, then one has to check 𝐺×𝐻² anomalies, where 𝐻 is one of the SM gauge groups.
On may check them for academic purposes, not consistency, ...
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Do different bases of Fock space commute?
For a given decomposition of the Fock space into $n$-particle subspaces, creation operator commute. For a $1$ particle state $|\psi\rangle$, there is an associated annihilation operator $a(\psi)$. If $...
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