16 votes

Are all fields in the universe we know of quantum fields?

Currently all fundamental fields are quantum, except for gravity. For this reason Quantum Gravity is a hot area of research, but the full Quantum Gravity theory has not been developed yet. Why not? ...
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16 votes
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How does canonical quantization work with Grassmann variables?

When you are dealing with Grassmann numbers you have a "left derivative" and a "right" derivative. A left derivative removes the variable from the left, a right derivative removes it from the right. ...
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  • 1,352
13 votes

Why can't $p^0$ change sign under a proper orthochronous Lorentz transformation?

Sometimes, diagrams are more useful than equations! Note that $p^2 = - (p^0)^2 + |\vec{p}|^2$ remains unchanged under any Lorentz transformation. Suppose that $p^2 < 0$ (say $p^2 = -1$). A plot of ...
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  • 20.9k
11 votes

Invariance of Action vs. Lagrangian in Noether's theorem?

No, they are not the same. To see why, even in classical mechanics, suppose we have symmetry transformation $q \rightarrow q + \epsilon K$ that leaves the Lagrangian invariant. This means that we must ...
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10 votes
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Classical field limit of the electron quantum field

The classical limit of bosonic quantum mechanical systems with both finite and infinite degrees of freedom is pretty well understood from a mathematical standpoint (with complete rigour, and for quite ...
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  • 11.4k
10 votes
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What is the actual form of Noether current in field theory?

Eq. (5) is (up to factors of the infinitesimal parameter $\varepsilon$) the standard expression for the full Noether current. Here: $\delta x^{\mu}$ is the so-called horizontal component of the ...
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  • 169k
10 votes

Why is action a functional of $q$ only?

The notation $q$ in the functional $S[q]$ stands for the whole parametrized curve/path $q:[t_i,t_f]\to \mathbb{R}$, not just a single position. In particular, the parametrized path already carries all ...
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  • 169k
10 votes
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Hamiltonian Field Theory in Peskin & Schroeder

When you transform from the Lagrangian to the Hamiltonian picture, you necessarily must choose a particular foliation of spacetime -- that is, you must single out a particular time direction, and ...
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  • 1,849
10 votes
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What is "gradient energy" in classical field theory?

The best intuition is from the example of a "loaded string": a set of $N$ masses that are connected by a massless elastic string. If we imagine the limit as $N \to \infty$ but with the ...
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9 votes
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Euclidean geometry in non-inertial frame

What is wrong is the idea that one can actually make the disk rotate; and it will remain perfectly rigid. In reality, what this correct argument shows is that relativity doesn't admit the existence ...
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9 votes
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Why is a nonzero VEV for a spinor field said to break Lorentz invariance?

The $v$ you write is itself a spinor, not a scalar. A non-zero spinor is obviously not invariant under Lorentz transformations, so a non-zero spinorial VEV breaks Lorentz invariance of the 1-point ...
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  • 107k
9 votes

Klein gordon field and positive/negative energy solutions

You say you're doing classical field theory, but the terminology comes from QM: these terms are only positive and negative energy if you interpret the field $\phi$ as a wavefunction, as people did ...
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  • 25.7k
9 votes
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Are fixed points of RG evolution really scale-invariant?

No, dimensionful couplings do not have to be all set to zero at an RG fixed point. An RG fixed point is one where all of the beta functions vanish, and beta functions generally have the form $$\beta(...
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9 votes
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Can Einstein-Hilbert action be derived from symmetry considerations?

Well, one can reason as follows: One wants a diffeomorphism-invariant action, which must be of the form $$ S=\int d^4x\sqrt{-g}L, $$ where $L$ is a scalar in terms of transformation properties. ...
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9 votes
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When can we handle a quantum field like a classical field?

A clean way to make the concept of a classical field precise it to phrase things in terms of a quantum effective action: given a generating function of connected and renormalised Green functions, $W(J)...
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  • 1,744
8 votes
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Counting Degrees of Freedom in Field Theories

You really should split your question. I will answer the part where you do not understand how counting of degrees of freedom work. Basically we count the number of propagating (physical) degrees of ...
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8 votes

Euclidean geometry in non-inertial frame

No, it doesn't violate the rules of geometry, it violates the rules of Euclidean geometry. Simple conclusion: for an observer fixed to a disk rotating uniformly relative to an inertial frame, the ...
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8 votes
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What is the point of complex fields in classical field theory?

Two real scalar fields $\phi_1$ and $\phi_2$ satisfying an $SO(2)$ symmetry and one complex scalar field $\psi$ are equivalent. However, the latter is more convenient because the particles made by $\...
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8 votes

Motivation for covariant phase space

Given a dynamical system $\dot x=F(x)$ with continuously differentiable $F$, there is a canonical bijection between initial conditions at a fixed time and solution trajectories. If the dynamical ...
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8 votes
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Scalar Field Theory for Gravity

If we linearize the equations of motion about a $\phi = 0$ background (and assume the usual sort of kinetic term), we will find a non-relativistic limit of something like $\nabla^2 \phi \propto \rho^2$...
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8 votes

Transformation of $d^4x$ under translation disregarded?

Perhaps the most clear way to see what's going on is to compare the action $$ S_1[\phi]=\int_{\mathbb R^d} \phi(x)^2\mathrm dx\tag1 $$ to the action $$ S_2[\phi]=\int_{\mathbb R^d} x^2\phi(x)^2\mathrm ...
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8 votes
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Uniqueness of the definition of Noether current

In Noether's first theorem, the continuity equation$^1$ $$ d_{\mu} J^{\mu}~\approx~0 \tag{*}$$ is an on-shell equation, i.e. it holds if the EOMs [= Euler-Lagrange (EL) equations] are satisfied. It ...
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  • 169k
7 votes
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Charged particle as observed from an inertial and a non-inertial frame of reference

First, I'll note that unless the non-inertial frame has a changing acceleration, there is some doubt as to whether it radiates at all. https://en.wikipedia.org/wiki/...
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7 votes
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What is a gauge theory?

A gauge symmetry is simply a symmetry transformation of the action that depends non-trivially on spacetime. You can ask for all physical theories whether such transformations exist. For the case of ...
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  • 107k
7 votes
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Most general Lagrangian in CFT in 0+1D

Classically a theory is invariant under a transformation if its action is invariant (up to boundary terms). In our case a conformal transformation is given by $$t'=\lambda t\\ Q'=\lambda ^{-\Delta}Q $$...
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  • 390
7 votes
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Theory invariance after substitution of theory's field equations back into theory's action functional?

TL;DR: Generically$^1$ an action principle gets destroyed if we apply EOMs in the action. Examples: This is particularly clear if we try to vary wrt. a dynamical variable that no longer appears in ...
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  • 169k
7 votes
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Is the Four-gradient of a Scalar Field a Four-Vector?

The $4$-gradient is a $4$- vector. Formally, when $x^\mu\to x'^\mu=\Lambda^\mu{}_\nu x^\nu$ $$ \begin{align*} \partial'_\mu &=\frac{\partial}{\partial x'^\mu}\\ &=\frac{\partial}{\partial (\...
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  • 2,483
7 votes
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When is Schwartz's method for "integrating out" a field valid?

I find that Schwartz is using the laziest way to discuss the integration of a field. What is going on can be made much more transparent rather easily. To simplify the discussion, I will assume that ...
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  • 10.9k
7 votes

Hamiltonian Field Theory in Peskin & Schroeder

While this is a good book, your question precisely touches on a serious issue: while in equation (2.5) their $\phi$ satisfies $$\phi(\mathbf{x}) = \phi(\mathbf{x},t),$$ in equation (2.25) their $\hat{\...
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