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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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Poincare invariant Lagrangian?

You are not mistaken, but compared to Lorentz invariance, translation invariance is usually trivial because there are usually no fields with non-trivial behavior under translations - you have $\phi(x)\...
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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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Sources to learn Gauge Theory, Groups, Lie Algebra, etc

Any “Mathematics for Physicists” at the suitable level will have an introduction to these things. In addition, most QFT books at the graduate level should have self contained introductions to these ...
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Parity in Effective Lagrangians

Perhaps a better question would be, what changes for a ${1\over 2}^+$ and a ${1\over 2}^-$ spinor. Likewise a $0^+$ and a $0^−$ spin-zero field. I frankly don't see your point: You multiply the ...
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Tachyon velocity in relation to light speed

This is like asking "how fast is a massive particle ?". The answer is "it depends". It depends on the reference frame / on the dynamics of the particle. Depending on the reference ...
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1 vote

Relativistic Euler-Lagrange equations for a four-vector (or one-form) field

It may matter if the metric components $g_{\nu\lambda}=g_{\nu\lambda}(x)$ [that we use to raise and lower indices with] depend on the spacetime coordinate $x^{\mu}$. This happens e.g. in GR. In this ...
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Question about the parity violation of weak interaction Lagrangian

$$ \gamma^0 P_L \gamma^0 = P_R, \\ \gamma^0 P_R \gamma^0 = P_L, $$ $$ P: \qquad \psi(x) \longrightarrow \gamma^0 \psi(-x) ~~~~\leadsto \\ P: \qquad \psi(x)^\dagger \longrightarrow \psi(-x)^\dagger ...
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What does it exactly mean by right and left functional derivatives?

TL;DR: A left derivative means a derivative that acts from the left. A right derivative means a derivative that acts from the right. In more detail, an infinitesimal variation of a functional $F[\phi]$...
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Problem 6 of Sheet 1 - Quantum field theory David Tong - Variation of Lagrangian density

In the last equation in your question, the term involving $\partial {\mathcal L}/{\partial x^\mu}$ is zero becuase ${\mathcal L}$ does not depend explicitly on $x^\mu$. Assuming that the other term ...
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2 votes

Can we write the effective field theory for the toric code model?

No. Effective field theory only describes system near critical point. Toric code model is far from critical point. Thus "No". Toric code model realizes a $Z_2$-topological order. When a ...
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Problem 6 of Sheet 1 - Quantum field theory David Tong - Variation of Lagrangian density

You can use the fact that $\mathcal{L}$, is a scalar field. Thus $\delta \mathcal{L}$ is of the same form as $\delta \phi$ just by replacing $\phi \rightarrow \mathcal{L}$. We further use the fact ...
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Noether current associated with transformation $\delta \psi=i\alpha \psi$

Mindful of @Robbie's comment, let's first rewrite your calculations as $$\begin{align}0&=\partial_\nu\frac{\partial\mathcal{L}}{\partial\partial_\nu\psi}-\frac{\partial\mathcal{L}}{\partial\psi}\\&...
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Relative signs between interaction terms

They determine the shape of the potential and hence its minima; possible implications include spontaneous symmetry breaking.
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Chiral symmetry of the Dirac Lagrangian

The already used identity for $\gamma^0$ can be generalized to arbitrary $\mu$, i.e. $[\gamma^\mu, \gamma^5] =0$, hence one picks up a sign in $$\gamma^\mu e^{i\alpha \gamma^5} = e^{-i\alpha \gamma^5}...
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How does the boundary term matter in scalar field and in more general cases?

Note first of all that it is usually important to specify appropriate boundary conditions (BCs) to render a variational principle well-posed, i.e. to ensure that the functional/variational derivative (...
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2 votes

Yet more gauge group nonsense: $D3$? $Q8$? $Z8$?

Big fan of gauge group nonsense. These finite group gauge theories are topological, so they don't have gauge bosons in the same way theories like electromagnetism do. They make for some interesting ...
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11 votes

What is the evidence that gravitational fields don't sum up as a superposition?

Black hole solutions would not exist in a linear theory of gravity. This is because black holes are vacuum solutions, not sourced by any matter, and there are no static vacuum solutions that die off ...
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4 votes

What is the evidence that gravitational fields don't sum up as a superposition?

Gravitational wave (GW) observations of binary black holes (BH) may provide experimental tests of superposition of spacetimes, as defined by you. Each BH is described by a spacetime metric, but the ...
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6 votes

What is the evidence that gravitational fields don't sum up as a superposition?

Einstein's equations are $$ G_{\mu\nu}[g] = R_{\mu\nu}[g] - \frac{1}{2} g_{\mu\nu}R[g] = 8\pi G_N T_{\mu\nu} \tag{1}. $$ where $g_{\mu\nu}$ is the metric of the spacetime. The Ricci scalar is given by ...
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1 vote

Discretization of derivative of delta function and affine Kac-Moody algebra

Let $\varphi$ be a smooth compactly supported function. Then : \begin{align} \int \varphi(x)\delta'(x)\text dx &= -\varphi'(0) \\ &=\lim_{\Delta\to 0} -\frac{1}{\Delta}(\varphi(\Delta)-\varphi(...
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Can a primary constraint contain spatial derivative of the field?

Yes, in field theory the primary constraints can contain spatial$^1$ derivatives of the canonical fields. One example is when Legendre transforming the Nambu-Goto (NG) string action, see e.g. this ...
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2 votes
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How is this Fourier transform done?

The first expression isn't a Fourier transform of a general field $\phi(x, t)$. I prefer to think of it as a mode expansion of a solution of the Klein-Gordon equation. (I explain why I say this a ...
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2 votes
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Four-vector differentiation (E-M Euler-Lagrange eq.)

As mentioned in the OP's link Schwartz (in his QFT book) doesn't keep track of the index placement on tensor objects that might obscure the structure a little. However, ignoring the first derivative $\...
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9 votes
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What does "conformally coupled scalar" mean?

A minimally coupled free scalar field is described by the action $$ S[g,\phi] = \frac{1}{2} \int d^D x \sqrt{g} g^{ab} \partial_a \phi \partial_b \phi . $$ However, this theory is not conformally ...
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What is the electric field and potential inside and outside grounded conducting and non grounded conducting sphere?

The conducting sphere is a Faraday cage: The field inside it is zero. What does the grounding do for that? Nothing! Remember that the electric field is the gradient of the potential and does not ...
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What is the correct way of looking at the Dirac field?

To add to what hft said, $\hat \psi$ is valued in a tensor product of operator densities and the spinor bundle. If you want to get an operator, you need to pair it with a smooth section of the dual ...
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What is the correct way of looking at the Dirac field?

I agree that both are operators but $\hat{\psi}$ is more than an operator. It is also a matrix. What is the correct way to think about $\hat{\psi}$ so as to distinguish it from $\hat{\phi}$? For ...
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4 votes

Must all field theories depend on the spatial derivate of the fields?

Spatial derivatives allow us to encode local interactions between the value of the field at different points. Ultimately, wavelike solutions to the equations of motion which propagate information from ...
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4 votes

Must all field theories depend on the spatial derivate of the fields?

If a field theory does not depend on the spatial derivatives of any of the fields involved, then the equations of motion will only depend on temporal derivatives; its form will be something like $$ \...
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1 vote

What does it mean by spin 1/2 or spin 2 field?

Relativistic fields are classified by how they transform under the Lorentz group. This means they are classified by representations of the universal cover of the Lorentz group ${\rm SL}(2,\mathbb{C})$....
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What does it mean by spin 1/2 or spin 2 field?

In geometrical view, spin $s$ particle returns its original state when it is rotated as $2\pi / s$. For example, spin $1/2$ has minus sign with rotating as $2\pi$, but it returns original state with ...
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