14 votes
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How to avoid paradoxes about time-ordering operation?

Time ordering $T$ (like any other operator ordering, such as, normal ordering $: ~:$, radial ordering ${\cal R}$, etc.) is technically speaking a linear map from symbols to operators, not a linear map ...
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  • 172k
12 votes

What are the adequate Hilbert spaces for Schrödinger, Schrödinger–Pauli, Dirac equations, and QFT?

Ordinary QM has essentially a fixed Hilbert space of $L^2(\mathbb{R}^n)\otimes S$, where $n$ is the number of spatial dimensions and $S$ some representation of the rotation group. This is due to the ...
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  • 108k
10 votes
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What's the importance of all four fundamental forces being "curvature"?

When we study non-gravitational fundamental interactions, we distinguish internal symmetries associated with only such interactions from the external symmetries of spacetime. For all fundamental ...
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  • 22.4k
9 votes

Euler-Maclaurin formula for Casimir Effect

Trick 1: Rewrite the integral so it has the bounds you want. \begin{equation} \int_0^\infty \nu d \nu = \int_0^1\nu d \nu + \int_1^\infty \nu d \nu = \frac{1}{2} + \int_1^\infty \nu d \nu \end{...
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8 votes
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Is it just a mnemonic to call $\phi (x)|0\rangle$ a particle at position $x$?

$\phi(x)|0\rangle$ is not the state of a particle (I stress that $\phi(x)|0\rangle$ is a one-particle state since I am referring to a free field) with position $x$ (when the temporal component of $x$ ...
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8 votes

Euler-Maclaurin formula for Casimir Effect

I want to add something to @Andrews answer. They gave a good answer which is technically correct, but I think there should be at least some things added to Trick 3. In the present answer there are ...
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7 votes

Weinberg, Effective Field Theories

Unitarity says that the sum of the probabilities for all possible out states (for a given in state) must be 1. Since the probability is the square of an amplitude, if any individual amplitude has a ...
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6 votes

BRST Symmetry and Single Particle States

It isn't obvious from that transformation alone. Remember that in P&S, the forward and backward polarization vectors are defined as: \begin{equation} \epsilon ^{\pm}_{\mu} = \frac{1}{\sqrt{2} |\...
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  • 1,051
5 votes
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BRST Symmetry and Single Particle States

I will post an answer because I have understood things in a certain way and I would like to share it. In this way, if it is wrong, I will get to know why it is wrong (if someone is kind enough to ...
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  • 1,587
5 votes
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Srednicki's QFT: Why $\langle p|\phi(0)|0\rangle$ in the interacting theory is Lorentz invariant?

A function is Lorentz invariant if $f(p) = f(\Lambda p)$. Consider the function $$ f(p) = \langle p| \phi(0) | 0 \rangle = \langle p| U(\Lambda)^{-1} U(\Lambda) \phi(0) U(\Lambda)^{-1} U(\Lambda) | 0 \...
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  • 21.4k
4 votes

Are the sources in QFT just particles?

Not really particles. The source terms $J$ are a computational tool. First, they allow you to take the functional Fourier transform of the path phase factor $\exp(i S[\phi] / \hbar)$. Why is this ...
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4 votes
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Does QFT Imply Strings the Same Way that String Theory Implies Branes?

When we take a good hard look at QFT, we don't see any particles (dimensionless points traveling on world lines). All we see are localized interactions. When we do our experiments to confirm these ...
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4 votes
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How is it justified to use two different coupling constants for tree-level diagrams and diverging diagrams?

I assume/hope the following post should clarify the issue: I'm missing the point of renormalization in QFT. We don't use two different coupling constants, we use the same constant. The perturbative ...
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4 votes

Which is the simplest QFT that describes the interaction between electrons and photons?

If you want to work in 3+1 dimensions where electrons are fermions, QED is the simplest. You can get scalar QED by giving up the latter condition. But if your goal is to become better acquainted with ...
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  • 5,004
4 votes

How to avoid paradoxes about time-ordering operation?

I think it's best to regard the time-ordering symbol as notation, rather than as an operator. When you see the time ordering symbol in time-ordered correlation functions, for example $\langle 0 | T \...
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  • 2,004
4 votes

Is the Uncertainty Principle valid for a macroscopic object at rest?

This is too long for a comment but there are lots of issues with your question. What do you mean by "car at rest"? The car is a composite object and the atoms in the material are certainly ...
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  • 39.5k
3 votes

What are the adequate Hilbert spaces for Schrödinger, Schrödinger–Pauli, Dirac equations, and QFT?

One way to proceed is to start with the classical observable and find some suitable Lie algebra containing these as generators. Upon quantization, the observables are promoted to operators, now ...
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3 votes
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Equal-time Canonical Commutation Relation for a scalar field

You can write the integrals in question as $$\int_{\mathbb{R}^3} f(\vec{p}) d^3 p = \int_{-\infty}^\infty\!\!\int_{-\infty}^\infty\!\!\int_{-\infty}^\infty\!\! f(p_x,p_y,p_z) dp_x dp_y,dp_z$$ where ...
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3 votes
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Which is the correct way to write a Lorentz group component in exponential form?

You are writing the same thing in different ways. You can indeed write a general Lorentz transformation in the form of $$\Lambda = e^A e^B \cdots,$$ where each term corresponds to a "simple" ...
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3 votes
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Delta function squared in Weigand's QFT notes

The square of the Dirac delta distribution does not make mathematical sense, cf. e.g. this Phys.SE post. However, physicists often implicitly imply that the spacetime is a finite (large) box. Then ...
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  • 172k
3 votes

In what way is String Theory "two-dimensional"?

A string worldsheet is 2D. One space dimension (how far you are along the string), one time dimension.
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3 votes
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Wightman Green function derivation

The $\epsilon$ in (3.59) does not shift the poles but rather regulates the integral over 3-momentum. More concretely, we start with (note the extra $i$ in fig. 3) $$ iD^+(t,\vec x) = \frac{1}{(2\pi)^4}...
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  • 231
3 votes

Source of spontaneous symmetry breaking

What you need for spontaneous symmetry breaking - and not just in QFT - is an unstable equilibrium, i.e. a potential maximum. The term $-\mu^2 \phi^2$ is the most simple expression for such a ...
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  • 1,295
3 votes

Muon pair production

You can answer this question by drawing the diagrams for the Mandelstam variables. Below, from Wikipedia, with time increasing to the right: In the s-channel, the intermediate trajectory is timelike, ...
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  • 73.9k
3 votes

How to avoid paradoxes about time-ordering operation?

$$ \mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t' \hat{H}_I(t') \right] =... $$ Is that just a definition? Seems like it must be, yes. Or perhaps better thought of as ...
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  • 8,436
3 votes
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Green Function expressed in terms of Hankel function (of the second kind)

There is a useful class of dummy integration variable trick, which applies also later when calculating Feynman diagrams. You can make the following observation: $$ \frac{1}{x} = \int_0^{+\infty}dt e^{-...
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  • 2,355
3 votes

Why 't Hooft says: field configuration in Euclidean space that have the vacuum (or a gauge transformation thereof) at the boundary

By `vacuum' he means a vanishing gauge field $A$ and by gauge transformation thereof he means some arbitrary $A$ which is related to the vanishing configuration by a non-Abelian gauge transformation. ...
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3 votes
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Transverse and longitudinal component of a photon propagator

For example, the paragraph below (12.56) $ D^{\mu v}(q)=D(q)(g^{\mu v}-\frac{q^\mu q^{v}}{q^2})+\frac{-i}{q^2} \frac{q^\mu q^v}{q^2}$says that the first term here corresponding to the transverse ...
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  • 8,436
3 votes

Relationship between Wilson's RG and the Callan-Symanzik Equation's normalization scale

Clearly, Wilson's approach and perturbative RG based on dimensional regularization are NOT the same thing. In Wilson's approach, you use a UV cut off and your flow equations explicitly depend on it. ...
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  • 1,155
2 votes

Why implementing $\partial_\mu A^\mu=0$ as an operator equation is not valid?

If you write out the functional dependencies in the equation you are referring to, you might see this more clearly. Note that $$ [\partial_i A^i(0,\mathbf{x}), A^\nu(0,\mathbf{y})] = \partial_{x,i} [A^...
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