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S-matrix for $\phi^3$ theory

It vanishes because of a hat in phi.
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1 vote

Polarization vector basis in Peskin & Schroeder

The transverse polarisation vectors are of the form $\varepsilon^{\text{T}}(0,\boldsymbol{p})$ with $\boldsymbol{k}\cdot \boldsymbol{p}=0$ and so $$\varepsilon^\pm \cdot \varepsilon^{\text{T}} \propto ...
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The relation between full Green's function and S-matrix

Condensed matter and QFT are interested in different quantities of the many-body systems they study (at least, on the level of basic textbooks). QFT aims at calculating the cross-sections of ...
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Mass terms for scalar lagrangians?

I'm a little rusty on this (since 2008, it's 2022) and I am not an expert, but a vector boson should have a field $A^\mu$ or $G_a^\mu$ depending whether it's a photon or a gauge boson, where the $\mu$ ...
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Difference between regularization and renormalization?

It's exactly like you said. Regularization is a mathematical procedure, with it, you can separate and isolate the divergence of the integrals. Regularization techniques include: dimensional ...
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Propagation of a wavefunction on a Riemannian sigma model

The configuration space for this theory has even coordinates $x$ and odd coordinates $\psi$ (it is $\Pi TX$ as a supermanifold). The Hilbert space is (the $L^2$ completion of ) the space of function ...
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2 votes
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How to second-quantize an operator if the field operator is a spinor

Let $\mathfrak h $ be the $1$-particle Hilbert space. Let $\mathcal O$ be an operator on $\mathfrak h$. The second-quantized version of this operator acts on an $k$-particle state $S_\nu(\phi_1\otimes\...
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2 votes
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Is CPT symmetry a direct consequence of Special Relativity?

Yes. Here is a sketch of how it works, although it can be proven under general assumptions. A Lorentz transformation takes the form \begin{bmatrix} \cosh{y} & 0 & 0 & \sinh{y} \\ 0 &...
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EFT's $\hbar$ counting at loop level

Ref. [1] is considering an UV action $S_{UV}[\phi,H]$ with no explicit $\hbar$-dependence with a cubic interaction $g_3\phi^2 H$ and a quartic interaction $g_4\phi^2 H^2$, and an EFT action $S_{EFT}[...
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8 votes
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Currently self-studying QFT and The Standard Model by Schwartz and I'm stuck at equation 1.5 in Part 1 regarding black-body radiation

It just a clever use of the geometrical series: $$\frac{1}{1-q} = \sum_{j=0}^\infty q^j$$ which is valid for any real number $q<1$. Here $q = e^{-E_n\beta} \equiv e^{-\hbar\omega_n\beta}$. As long ...
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1 vote

EFT matching: using tree-level to perform 1-loop-level

Apparently, the answer lies in the fact that, in equation $(2.29)$, one has the scaling masses, dependent on $\mu$. Therefore the terms in $\log(\mu^2/m^2)$ aren't canceled, but are included in the ...
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-1 votes

What are zero-energy Rindler photons?

Although these "zero-energy" Rindler photons carry no longitudinal momentum, it's interesting to note that they carry transversal momentum. In this case, I think, the vanishment of the ...
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4 votes
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What is the origin of these log terms in dimensional regularization?

These kinds of logs appear whenever you have things like $a^\epsilon$ where $\epsilon$ is a small parameter we are expanding in powers of. In fact, by definition $a^\epsilon = \exp(\epsilon \log a)$ ...
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3 votes
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Antifields in BV formalism - do they also have gauge transformation laws?

Well, OP's title question is partially a matter of conventions: Usually one only assigns gauge transformations to the original field sector of a gauge theory. E.g. in Yang-Mills theory, that would be ...
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Are the particles in the zero-energy Higgs field real or virtual?

Do not confuse the concept of a field in the quantum field theory of particle physics, the standard model, with the particles that creation and annihilation operators create and annihilate on those ...
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Gell-Mann Low formula vs time independent perturbation

The main reason modern textbooks introduce the Gell-Mann-Low formula is that it leads to a very simple proof of the Feynman rules. The argument roughly goes like this: first, we write $$ U(t_1,t_2)=\...
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Dimensional regularization vs. hard cutoff and their relation to the renormalization scale in 2d vs 4d to find $\beta$ functions

Here's a rough argument for why the latter part of your question going from a pole to beta function is valid. (As I mentioned in the comments, the use of an additional scale $M$ in addition to $\mu$ ...
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Expectation values in path integral formalism

On one hand, the formal connection between the operator formalism in the Heisenberg picture and the Hamiltonian phase space path integral is $$\begin{align} &{}_J\langle Q_f,t_f |TF[\hat{Q},\hat{...
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Why is $\mathcal{M}(k)$ given by this? (Ward Identity derivation in Peskin & Schroeder)

It is due to the simplest application of relativistic perturbation theory. The S-matrix is defined with $H_I$ as interaction Hamiltonian $$S= T\exp\left(-i \int_{-\infty}^\infty \hat{H_I} dt \right)$$ ...
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5 votes
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Does the neutrino interact with the photon?

Yes, the neutrino may have a magnetic moment at the 1 loop level in vacuum, cf here, e.g. This is summarized by your unrenormalizable effective (fake tree) dimension 5 operator, since the loop into ...
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About $\phi^4$ theory

Why in the interaction term of the hamiltonian operator do we encounter the free field? Because we work in the Interaction picture. You stated this in your question, so I think you already know about ...
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How are we able to use quantum field theory to study systems?

How are we able to use quantum field theory to study systems? The same question could also be asked about quantum mechanics, or even about classical mechanics. For example, in the Newtonian mechanics ...
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Does the vacuum really have infinite energy density?

The energy of the vacuum is also known as "dark energy", or the cosmological constant. The value of this in Planck units is about $10^{-120}$, so not only is it non-infinite it is nearly ...
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Does the vacuum really have infinite energy density?

Indeed the present theory predicts a very high or diverging vacuum energy of the universe. This is known the cosmological constant problem or vacuum catastrophe. It is a consequence of the zero point ...
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1 vote
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Approximating high-energy Compton scattering cross section

\begin{equation} \frac{d\sigma}{d\cos\theta} = \frac{\alpha^2\pi}{2} \frac{1}{E(E+\omega\cos\theta)} \end{equation} Look at the denominator. Start by writing $E(E+\omega \cos\theta) = E\omega(E/\...
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4 votes

Does the vacuum really have infinite energy density?

Quantum field theory does not say that "the vacuum has infinite energy density" because part of our modern conception of quantum field theory is the idea of renormalization. Renormalization ...
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Explicit check of Ward identity (Peskin & Schroeder p. 160)

You can further simplify this expression by using the dirac equation $$ 0=(\not p-m)u(p)=\bar u(p')(\not p'-m) $$ and $k+p=k'+p'$. Then the second term can be expressed as $$ 2\not k p^\mu-\not k\not ...
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Particle-hole symmetry in 2nd quantization

This is a good question that I have also had, so I'm answering this even though it is two years old! It looks like your definitions are all fine, but I would use \begin{equation} \hat{\mathcal{P}}\...
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Dirac propagator in Non-Abelian Theory

Hints: P&S is considering the free fermion propagator in eq. (16.4), i.e. the cubic $\bar{\psi} A\psi$ interaction term does not contribute. In the Lagrangian density (16.1) there are implicitly ...
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Free energy from the vacuum?

As per energy conservation law (or "no free lunch theorem"),- whatever energy you'll put into a pair production - on annihilation it will return given energy back to the field, so that total ...
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7 votes
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How are we able to use quantum field theory to study systems?

That is an accurate summary of the situation, yes: the Hilbert space cannot be split into a tensor product of system and environment, for the reasons that Witten explains. Briefly put, the reason ...
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2 votes

Wick's Theorem and Functional Derivative

The 1st equality in eq. (5.155) follows from the definition of an Euclidean correlator function $$\begin{align} \langle F[\phi] \rangle_J~=~&\frac{1}{Z[J]}\int\!{\cal D}\phi~ F[\phi] \exp\left\{ -...
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Wick's theorem: From operators to fields

I assume you mean a field operator rather that a field. Just expand the field operator in modes and the coefficients are your operators on which you can use Wick’s theorem.
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Parke-Taylor formula and MHW-amplitudes

As a convention, all the gluons are outgoing. One can interpret $M[1^- 2^- 3^+ 4^+ 5^+]$ as $3^- 4^- \rightarrow 1^- 2^- 5^+$. This also relates to why these class of amplitudes are called MHV, ...
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What is the role of Hermitian Hamiltonians in relativistic QFT?

Things happen in Fock space in QFT, but I don't remember seeing any result looking like total probability equals to 1 over Fock space (does such a sum even converge?) However, the closest result I ...
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What is the role of Hermitian Hamiltonians in relativistic QFT?

In Q field theory we are quantizing the fields not the particles. In some (but not all) field theories there are excitations that look like particles, but their conservation or lack of has nothing to ...
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2 votes
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Can quantum fields be artificially excited?

It first needs to be clear what you understand by "artificially" and "naturally". The LCH (as well as others particles accelerators) creates the conditions to produce particles (...
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Can quantum fields be artificially excited?

You can "artificially" create excitations of a quantum field in your own home. Just flip on a light bulb and you will generate trillions of excitations in the photon field. Lasers are a ...
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Free energy from the vacuum?

If the amount of charge on the electrodes producing the electric field doesn't change then I don't see how the electric field can lose energy in this process. Total EM energy (in the static case) ...
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1 vote
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Choosing the polarization vectors, Quantization of EM field

I think you can define the polarization vectors $\mathbf e_{\mathbf k}^{(1)}, \mathbf e_{\mathbf k}^{(2)}$ for the given $\mathbf k$ in any way, provided their dot product is zero. There is no need ...
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Using Wick's Theorem in an example with the harmonic oscillator

I understand Wicks theorem to be, $$T(x)=\mathcal{N}(x)=\sum:\textbf{all contractions}:$$ No, the above is incorrect. The time ordering and normal ordering are not generally the same, which is why ...
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1 vote

A question about the Dirac operator and zero modes in the book "Mirror Symmetry" by Clay Institute

I don't know exactly what the book is doing, but in calculating correlators of Fermi fields interacting with a dynamical background gauge or gravitational field the Green functions of the Fermions ...
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

First, let's consider the derivative $\partial$ as an operator on the Hilbert space $\mathcal{H} = L^2(\mathbb{R}^n)$ of square-integrable functions on $\mathbb{R}^n$ (the space typically encountered ...
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3 votes
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Symmetry implies Ward identity

$U = e^{i\epsilon Q}$ is a symmetry if : $$\langle\alpha | e^{-i\epsilon Q}Se^{i\epsilon Q}|\beta\rangle = \langle \alpha|S|\beta\rangle \tag{1}$$ You can see this as saying that the $S$-matrix is ...
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Product of Lorentz invariant factors may be Lorentz non-invariant

I have an answer. After banging my head off of chain rule and change of variables non-sense all day, I recalled that 0+0=0 and $\frac{d}{dx}xy=y$. That means that the Lorentz test for one 4-vector ...
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11 votes

How are anomalies possible?

It's true as the others have said that the path integral measure may not be invariant. However I don't really like that quote as a general description of anomalies, since it sounds like they rely on a ...
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8 votes

How are anomalies possible?

The reason for this is that in the classical theory the Lagrangian fully specifies the dynamics of the system, however, in the Quantum system this is not true. Rather the Quantum theory is given by (...
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9 votes

How are anomalies possible?

This is such a short answer I'm tempted to just leave it as a comment, but besides specifying the Lagrangian you need to also specify your regularization, and the regularization is what introduces the ...
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Does the 4-vector gradient commute with "itself"? If yes, why do they commute?

For a smooth scalar functions $f$ (which are the objects on which the vector fields $\partial_\mu$ are defined to act), we simply have that $$[\partial_\mu,\partial_\nu]f=\partial_\mu\partial_\nu f -\...
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