New answers tagged quantum-field-theory
5
votes
Renormalization in quantum field theory by discretizing space (but not time)
Does this work (that is, are all the regularized integrals associated with Feynman diagrams actually finite in this scheme) or is there some problem I'm not seeing?
At least perturbatively (which is ...
0
votes
In QFT when performing path integral, why don’t we divide it by the volume of Poincaré group, as what we did for gauge group?
We should consider the generating function $Z[J]$ rather that just the partition function $Z = Z[0]$, because it computes the correlation functions which contain the physical content of the theory.
If ...
2
votes
Accepted
Time ordered correlator from path integral: equation of motion?
Let $P(\Phi)$ be a polynomial in a set of field variables $\Phi= \{\phi_1(x), \phi_2(x),\ldots\}$, and
consider a correlation function
$$
\langle{P(\Phi)}\rangle= \frac 1 Z \int d[\Phi] P(\Phi)e^{-S[\...
-1
votes
Problem with Bohr Frequency in Quantized Radiation - Matter interaction
The energy variation as per Dirac's Time-Dependent Perturbation Theory would be no other than field-as-atom's relativistic confounding of radiation gauge measurement with interference on the part of ...
0
votes
Operator-state correspondence in QFT
A small enhancement to the answer above, also meant as a answer to the comment of @LoganM.
In general you can do the same steps, you can do the coordinate transformation to $z,\bar{z}$, and then look ...
4
votes
Wilsonian RG in QFT: what is the difference between renormalized and bare couplings?
A main idea of the Wilsonian renormalization group (RG) flow is, as OP writes, that the renormalized coupling constants $g$ become the UV bare coupling constants $g_0$ when the renormalization scale $\...
4
votes
Doppler shift of a single photon
If your photon has a wave vector $\vec k$ in the lab frame, the it has a covariant 4-vector in the lab:
$$ k^{\mu} = (\omega/c, \vec k) = (k, \vec k) $$
where I have used the dispersion relation:
$$ \...
3
votes
Doppler shift of a single photon
A single photon is a standard electromagnetic wave, just one of the minimum amplitude. The Doppler shift is the same as for electromagnetic waves of higher amplitudes.
6
votes
Accepted
Are Weinberg's soft theorems relevant when making predictions about collider physics?
Gravitons have never been observed, much less graviton scattering, so the graviton soft theorem is not relevant for particle physics experiments.
The graviton soft theorem has been shown to be related ...
0
votes
Can Special Relativity's Effect on Proton Diameter Explain Magnetic Fields?
I don't have time to watch the video, but I'm assuming that we are dealing with two frames here:
Frame 1: the protons are stationary, the electrons are in motion, and the wire is overall neutral
...
0
votes
Confusion with Weinberg's QFT book, Volume I, Equation 2.5.3 (one-particle states as irreps of Poincare group)
Here is my answer. Feel free to correct me.
Under an arbitrary proper, orthochronous Lorentz transformation, $\Lambda$, takes us from $p\to p^\prime=\Lambda p$. However, we have to consider the set of ...
0
votes
Confusion with Weinberg's QFT book, Volume I, Equation 2.5.3 (one-particle states as irreps of Poincare group)
Weinberg proves directly before the equations you mention that
$$P^\mu U(\Lambda)\Psi_{p, \sigma} = \Lambda^\mu{_\rho}p^\rho U(\Lambda)\Psi_{p, \sigma}.$$
In particular, $U(\Lambda)\Psi_{p, \sigma}$ ...
1
vote
Question on Majorana Path Integral
As Prahar stated in the comments, this is analogous to integration on the complex plane. More simply, if you are integrating on $\mathbb{R}^2$, you pick coordinates $(u, v)$ and construct a measure $\...
2
votes
A free parameter when switching from $\phi$ to $a$
It should be fine as long as you put the right normalization in $\pi$. If one takes $$\pi(x) = i\sqrt{\frac{m_x}{2}} (a_x^\dagger - a_x)$$
then everything works out.
In general, you can always add a ...
2
votes
Accepted
Expanding the generating functional $W[J]$ for connected diagrams as a power series in $\hbar$
OP is essentially asking about the $\hbar$/loop-expansion for the generating functional $W_c[J]$ of connected diagrams, i.e. that the power of $\hbar$ in a diagram is given by the number of loops. ...
1
vote
Accepted
Why are the corrections to the effective Lagrangian (Wilsonian renormalization) given by connected diagrams only?
The Wilsonian effective action is defined as
$$\begin{align} \exp&\left\{-\frac{1}{\hbar}W_c[J^H,\phi_L] \right\} \cr
~:=~& \int_{\Lambda_L\leq |k|\leq \Lambda_H} \! {\cal D}\frac{\phi_H}{\...
2
votes
Transmitting data via quantum fields
All fields in our universe are limited to the speed of light for propagation.
If you want long-range communication, you need a massless excitation. For it to be practical, you need an easy & cheap ...
0
votes
Weinberg QFT Vol 1, charge renormalization
One thing we said is wrong:
When we rewrite the term
$$
q_B \overline{\psi}^B A^B_\mu \gamma^\mu \psi^B
= Z_1 q_R \overline{\psi}^R A^R_\mu \gamma^\mu \psi^R
$$
where $A_B = \sqrt{Z_3}A_R$ and $A_R$ ...
1
vote
Accepted
Does matter in the outside universe affect Hawking radiation?
Hawking radiation emits all sorts of particles, including charged particles. Adding a big charge to the outside of the black hole would not alter, in principle, the particles emitted by the black hole,...
0
votes
In which systems does Planck's constant apply? Is everything thus quantized?
Most physical interactions we know of are currently described by quantum mechanical theories. At present, there are multiple candidates for a quantum theory of space and time, which would also be a ...
1
vote
Does the density matrix still holds for infinite subsytems?
I think the issue here is the assumption here is the assumption that there is always a well defined maximally mixed state. If we take the quantum states in the computational basis as a binary ...
1
vote
Accepted
In which systems does Planck's constant apply? Is everything thus quantized?
Of course, at the classical or macroscopic level, not all things are quantized, e.g. the energy of a ball rolling downhill or the amount of soup a person can eat in the morning. However, at the ...
4
votes
What, specifically is meant by "particles are popping in and out of existence all the time?"
For a less technical answer, things like vacuum fluctuations and virtual particles and so on are purely artefacts of a mathematical trick we use to simplify the problem.
When we want to know what ...
4
votes
What, specifically is meant by "particles are popping in and out of existence all the time?"
Thank you for this question. I've been waiting for an opportunity to address this issue. It is an important valid question.
The short answer is: there are no particles popping in and out of existence. ...
0
votes
How to obtain the explicit form of Green's function of the Klein-Gordon equation?
I was interested in this problem, specifically in computing the retarded Green's function, for which only the $\tau^2>0$ sector survives. One way you can do this is by first breaking down the ...
1
vote
Why reasonable observables are made of an even number of fermion fields?
One explanation is that on one hand an observable should be a quantity/operator that could take a value in a measurement in the form of (a tuple of) ordinary numbers.
On the other hand (an odd product ...
1
vote
If "borrowing energy for a short time" interpretation of HUP is wrong, then how are the ranges of fundamental forces explained?
I think the reason for resistance to the notion of "borrowing energy" is that in the Feynman method the energy and momentum is strictly conserved at every stage: in every vertex and in every ...
1
vote
Accepted
Charge renormalization choice in QED
This is common practice whenever dimensional regularization is used. Simply consider the dimensions of the various terms in $d$ space-time dimensions. As the action $S= \int d^d x \, \mathcal{L}$ is ...
3
votes
How can a QFT field act on particle states in Fock space?
The two sentences :
Excitations of the quantum fields are particles
and
The quantum fields acts on the multi-particles states in the Fock space
both encapsulate the same idea because excitations ...
2
votes
How can a QFT field act on particle states in Fock space?
In both quantum mechanics (QM) and quantum field theory (QFT) Heisenberg picture observables give us information about the evolution of measurable quantities. Observables don't directly represent the ...
2
votes
Accepted
Making sense of stationary phase method for the path integral
You want to compute the generating functional $$Z[J]=\int [d\phi] e^{\frac{i}{\hbar} (S[\phi]+\phi \cdot J)}, \quad Z[0]=1, \quad \phi\cdot J:=\int\! dx \, \phi(x) J(x)$$ in the quasiclassical ...
1
vote
Making sense of stationary phase method for the path integral
Your understanding seems fine (if we can deal with these questions it will be perfect).
To obtain (2) the picture I was taught is to see $\phi$ as a big vector with values in each point of space-time. ...
7
votes
How can a QFT field act on particle states in Fock space?
You had encountered exactly the same situation already in elementary quantum mechanics, but maybe you did not recognize the analogies. Consider a single harmonic oscillator with angular frequency $\...
5
votes
How can a QFT field act on particle states in Fock space?
The field operators do not represent particles, they are operators that, if acting on a state (a vector in fock space which represents a state with or without particles), can change the state of the ...
7
votes
Feynman propagator as a sum over eigenfunctions
We start from the defining equation
$$ (m^2+ \square_x -i \epsilon) G(x-x^\prime)=\delta^{(d)}(x-x^\prime) \tag{1} \label{1}$$ of the Green function of the Klein-Gordon operator in $d$ dimensions with ...
2
votes
Analytical continuation as regularization in Quantum Field Theory, the remaining questions
For your first question, analytical continuation is one way out of many ways to regulate a sum. As you pointed out, there is always an ambiguity in attributing a sum to a diverging series. Analytical ...
0
votes
Accepted
Quantum effective action for Yang-Mills theories
Your action seems identical to the effective action in section 2.4.2 of David Tong's notes on gauge theory (modulo the sign in front of $[F_{\mu \nu}, \cdotp]$).
It can be obtained using the ...
Community wiki
1
vote
Parity operator action on quantized Dirac field
I don't have my copy of the book to hand, but I assume that $P$ is there a linear operator on the Hilbert space. It does not commute with the creation and annihilation operators:
$$
P\hat a_{\bf k} ...
0
votes
The contradiction between Gell-mann Low theorem and the identity of Møller operator $H\Omega_{+}=\Omega_{+}H_0$
Actually, the original Gell-Mann-Low Theorem(GLT) can be deduced by the Adiabatic Theorem(AT), which requires no degeneracy and discrete spectrum.
However, in more general cases, say, degenerate ...
1
vote
Is conservation of energy a local law in Quantum field theory?
Yes the stress energy tensor operator, also in curved spacetime, satisfies the local conservation equation in a distributional sense $$\nabla_a :\hat{T}^{ab}:_{\omega}(x)=0\:,\tag{1}$$ where the &...
6
votes
Interpreting generating functional as sum of all diagrams
Perturbatively, one can formally argue that the partition function/path integral/functional integral/generating functional can be written as
$$\begin{align} Z[J]~=~&\int {\cal D}\frac{\phi}{\sqrt{\...
3
votes
Accepted
Amputated connected 2-point function is inverse to connected 2-point function
TL;DR: Use the relations $$D_2~=~\int E_1A_2E_2\tag{1}$$ and
$$E_1~=~D_2~=~E_2\tag{2}$$ to conclude OP's sought-for relation
$$A_2~=~D_2^{-1}.\tag{3}$$
Eq. (2) can be argued in at least 2 ways:
...
7
votes
Accepted
Interpreting generating functional as sum of all diagrams
The generating functional by definition is an object that encodes the Green's functions in a series as $$Z[J]=\sum_{n=0}^\infty \dfrac{1}{n!}\int dx_1\cdots dx_n J(x_1)\cdots J(x_n) \langle \phi(x_1)\...
0
votes
Are vacuum energy, zero point energy and vacuum fluctuations the same thing?
Vacuum energy and zero-point energy are the same thing. Vacuum fluctuations, however, do not exist. The vacuum state is completely stationary in time. The concept of vacuum fluctuations is from the ...
2
votes
Are vacuum energy, zero point energy and vacuum fluctuations the same thing?
Terminology-wise, vacuum energy and zero-point energy are the same thing, though they are used in slightly different contexts. Zero-point energy applies to the residual energy of any quantum ...
0
votes
How does parity act on relativistic one-particle states?
I think your problem depends on whether the particle is massless or not, that is, the standard momenta associated to massless particle and massive particle are different. And $\sigma$ plays different ...
1
vote
Accepted
Free acoustic phonon propagator
The reason why you get a weird pole at $\omega = 0$ in your first attempt to find the propagator of a free acoustic phonon is that the transversal oscillations which are always part of the ...
1
vote
Why does $S$-matrix theory end up being a covariant formalism when it is not obvious that it is?
The time ordered "$T$" product is formally Lorentz covariant because the fields commute at spacelike separation --- so "$t$" is in no way being singled out. I say "formally&...
1
vote
What particles are described by the Klein-Gordon Equation?
The Klein-Gordon equation is the field version of the special relativistic energy momentum relation. When one abandons the restriction that it should only be used for scalar fields it becomes clear ...
4
votes
Accepted
What particles are described by the Klein-Gordon Equation?
A real (hermitean) Klein-Gordon field describes chargeless spin $0$ bosons. Examples: $\pi^0$, Higgs.
A complex (non-hermitean) Klein-Gordon field describes charged spin $0$ bosons, where the "...
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