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Note that the spectrum ${\rm Spec}(\hat{A}) \subseteq \mathbb{R}$ of a Hermitian/self-adjoint operator $\hat{A}$ belongs to the real axis $\mathbb{R}\subseteq \mathbb{C}$, cf. e.g. this Phys.SE post. It is therefore not surprising that a reality condition for a classical field naturally translates into a Hermiticity/self-adjointness condition for the ...


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As other answers mention, it was originally (in QED) about getting a neutral vacuum. It is useful to go back to Schwinger's old version of QED, before Dyson's approach became accepted. See Pauli: Selected topics in field quantization. Pauli presents both ways of looking at it: 1) define the electric current as sum of two terms (p.20 [6.4]), such that the ...


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First of all, note that the radial operator ordering ${\cal R}$ is implicitly implied in many textbooks of CFT (e.g. Ref. 1). For instance, eq. (2.2.7) on p. 39 in Ref. 1 is discussing Wick's theorem between two operator ordering prescriptions. In this case between normal ordering $:~:$ and radial ordering ${\cal R}$. See also e.g. this Phys.SE post. The ...


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The choice of normal ordering prescription $:~:$ is typically adjusted to the choice of vacuum state $|\Omega\rangle$ so that the bra-ket-sandwich of normal-ordered operators $$\langle \Omega|:\hat{\cal O}_1\ldots \hat{\cal O}_n : |\Omega\rangle~=~0 $$ vanishes. The relation of normal ordering prescription to Wick theorem and other operator ordering ...


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In classical physics, quantities are ordinary, commuting $c$-numbers. The order in which we write terms in expressions is of no consequence. In quantum field theory (QFT), on the other hand, quantities are described by operators that, in general, don't commute. Classical physics is a low-energy approximation of quantum physics - the road from quantum to ...


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About CPT-conservation. If you talk about graviton then you talk about linearized GR. There is theory which tells us that all irreducible representations $\left(\frac{n}{2}, \frac{m}{2} \right) \oplus \left( \frac{m}{2}, \frac{n}{2}\right)$ of the Lorentz group is invariant under $C, P, T$ transformations. The theory of graviton (it may be defined without ...


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It seems there is a mistake in your $\pi (x)$ expression: there must be one minus sign near $\hat{a}^{\dagger}$. The relation between classical and field is obvious since lagrangian (hamiltonian) of free $\varphi $ (it's not hard to see that $\varphi$ satisfies Klein-Gordon equation) field may be rewritten as lagrangian of free ossilator in terms of ...


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The different string vibration modes represent particles. There are infinitely many different excitations of increasing mass and experiments usually have a limited energy scale E, so we don't worry about particles with m>E. One can then only consider e.g. massless modes and then build an effective theory describing their interactions. One can do this by ...


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If by asymptotic safety you mean theories that have an interacting UV fixed point, they're a dime a dozen in field theory. Just pick any conformal field theory you like and see if it contains a relevant scalar operator. If so, deform away from the fixed point with that operator. From the viewpoint of wherever it ends up in the IR, the UV is "asymptotically ...


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There are a number of preprints discussing this. I'll just list out the ones I've read: Sandor Nagy, "Lectures on renormalization and asymptotic safety". Annals of Physics 350 (2014), pp. 310-346. Eprint arXiv:1211.4151 DOI:10.1016/j.aop.2014.07.027 Jens Braun, Holger Gies, Daniel D. Scherer, "Asymptotic safety: a simple example". Phys.Rev.D 83 (2011) ...


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The Standard Model Lagrangian before and after spontaneous symmetry breaking (SSB) is renormalizable. To see that recall that the rule is (though it may not be immediately obvious as to why this rule holds) that a theory is renormalizable if all the terms in the Lagrangian are of dimension 4 or less. This is true by design for the Standard Model in which all ...


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Earlier answers seem to be off-topic, since their authours talk about real photons, while the question is asked about the picture of virtual photons which serve as interaction mediators for the electromagnetic interaction, even in the electrostatic case. The most important thing to settle beforehand is that the picture of interaction by exchanging virtual ...


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By trading photons? I'm not sure if I missed something everybody else knows, but an electron(-) stays near a proton(+) because opposites attract. Unless you meant, the photon emitted by an electron moving into a lower energy shell, interacting with the electrons of another atom. To answer some of your questions: Where does the energy to emit a photon ...


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I think you are confusing two different subjects. Let me explain, and I'll give you the simplified version. If you ask I'll update with a more advanced explanation. There are four fundamental forces in the universe. Strong Nuclear, Weak Nuclear, Electromagentic, and Gravity. Electrons and protons mainly interact by the electromagnetic force, since this ...


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In classical mechanics, it is often possible and convenient to describe a system with an object called a Lagrangian (in that it governs a system's behaviour, the Lagrangian is similar to a Hamiltonian). Like the Hamiltonian, the Lagrangian ought to be real - and any terms inside the Lagrangian ought to be Hermitian. In quantum field theory (QFT), the ...


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There are, in general, infinitely many operators with equally spaced eigenvalues. Suppose a self-adjoint operator $A$ has a purely discrete spectrum (i.e. it is either compact or with compact resolvent) and denote by $\{\lambda_i\}_{i\in\mathbb{I}}$ its real eigenvalues ($I\subseteq \mathbb{N}$): then by the spectral theorem it can be written (on its domain ...


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Dirac fermions is only the direct sum of left- and right-handed Weyl representations (which leads to time inversion, charge inversion and spatial inversion invariance of the theory). Two Weyl representations are mixed by the mass term in the Dirac equation. If we set mass to zero, we will get two uncoupled equations, each of which describes Weyl fermion. But ...


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Yes, the parity symmetry can be implemented only for the four-component Dirac fields and not for two-component Weyl fields. This fact, mathematically speaking, does not depend on the value of the mass. Physically speaking, however, I am not sure that a massless four component spinor makes much sense in standard theories (Sorry, I do not know anything about ...


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Is there any phenomena where the wave description of electron (motion) is not applicable? Let us touch base and have a look at this bubble chamber picture: This is one single elementary particle interaction, rare, because it is the creation of an omega particle, but it is one single instant. There is nothing wavelike in this datum. The particles ...


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Probably only the collapse of the electron wave function, which occurs point-like in experiments. That means, electrons make point-like response on photographic plates, CRT screens and in other instruments, both in spatial and temporal sense. No wave description of this process has been built (yet). Maybe also that several electrons make a many-particle ...


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The physics of particle mechanics is contained within quantum mechanics, so I don't know if it's correct to say that there is a place where the "wave description is not applicable." However, there are experiments where the wave description is not apparent. The beginning paragraphs of this page give a very short historical account of wave & particle ...


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Look at each vertex independently. The vertex including the photon also involves a hadron, so the exchange is going to be strong mediated. The eta carries certain quark flavors. These can-not have come from the photon, so they came from the exchange particle. You have to conserve angular momentum between the initial and final states, whic may involve a ...


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Indeed, the Standard Model is consistent in perturbative expansions, which means that we do not know if the Standard Model is consistent or not. So it is possible that the original Standard Model with 15 Weyl fermions per family is not consistent. In other words, there may not exist any well defined quantum model, whose low energy effective theory reproduce ...


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The Standard Model is consistent in perturbative expansions. It is inconsistent non-perturbatively but all these inconsistencies only show up "qualitatively" at energies well above the Planck energy – where we know the non-gravitational Standard Model to be inapplicable, anyway. The inconsistencies of the Standard Model involve the Landau poles – the ...


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For me, the most clear reference on the optical theorem and on Cutting rules is chapter 7.3 of Peskin & Schroeder.


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Have a look here, and here. cutting is essentially a shortcut for calculating complicated diagrams.


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A soliton is a localized, non-dispersive solution of a nonlinear theory in Euclidean space. It certainly is a real object: you have a famous story about a certain John Russell who observed soliton-like waves made by a boat on a river (wikipedia knows everything about it!) The so-called morning glory clouds in Australia ...


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It seems to me that the main argument given in the book is: while irreducible representations of CCR are all unitarily equivalent in finite dimensional QM, this is no more true in infinite dimensional QM. This leads, roughly speaking, to the necessity of choosing a specific Hilbert space (with a specific realization of the CCR) depending on the system ...


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It is very important to distinguish whether the symmetry is broken explicitly or spontaneously. I think that the sentence "Now when I break this symmetry spontaneously (or explicitly)" indicates that its author isn't quite distinguishing these things. An explicit symmetry breaking generally lifts the degeneracy because the different parts of the multiplets ...


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Valter's answer is completely correct, but I'll just briefly expand on it to address the specific values you ask about. The place to go, really, is the Wikipedia page Particular values of Riemann zeta function, which lists mosts of the values of $\zeta(s)$ (which, as Valter explained, equals $$\zeta(s) := \sum_{n=1}^{+\infty} \frac{1}{n^s}$$ when ...


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The true fact is the following. Consider $$\zeta(s) := \sum_{n=1}^{+\infty} \frac{1}{n^s} \quad \mbox{with $s\in \mathbb C$ and } Re \:s >1\:. \tag{1}$$ That function, with the said complex domain, is well defined (the series absolutely and uniformly converges) and is a complex analytic function. As a consequence of a well-known theorem on analytic ...


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Holographic renormalization for non-conformal branes, i.e. non-AdS/non-CFT systems, was systematically developed in this paper by Kanitscheider, Skenderis and Taylor. They even work it out for the example of the Witten model, which is the background of the Sakai-Sugimoto model. The key principle that allows one to extend the formalism of holographic ...


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In QM the Schrödinger equation, is the equivalent of Newton's law in Classical Mechanics. The Schrödinger equation describes the state of a quantum system (i.e. atoms, subatomic particles etc.), and how the quantum system changes over time. I think you are getting confused because there are two main places where the term wave appears. (1) The Double Slit ...


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Yes,the photoelectric effect can be explained without photons! One can read it in L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995, a standard reference for quantum optics. Sections 9.1-9.5 show that the electron field responds to a classical external electromagnetic radiation field by emitting ...


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The gauge connection is not unique, and this has nothing to do with the presence of matter fields. Let $\Sigma$ be our space-time, $P$ a principal $G$-bundle, and $\mathcal{A}$ the space of connections on $P$. Then, gauge transformations $t : P \to G$, forming the group of gauge transformations $\mathcal{G}$ have an action on $\mathcal{A}$ given by $$ A ...


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Of course @QMechanic's answer is correct. i would like to show a very simple reason why this is so (and also point to possible generalisations) First of all, any complex number $z=a+bi$, is 2-dimensional and each part (real $a$ or imaginary $b$) can be completely independent of each other. As a result a complex number can represent in condensed form 2 ...


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This is essentially the statement that the Fourier transform is injective and therefore invertible. In looser language, it states that the functions $\mathbf x\mapsto\exp(i\mathbf p\cdot\mathbf x)$ are linearly independent, so any sum of them (i.e. $\int\mathrm d^3\mathbf p$) that gives zero must have identically zero coefficients. Thus, it applies to any ...


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Your first equation can be compared to (in one dimension) $$ g(x)=\int dp\exp(ipx)f(p)=0 \,\,\, (*) $$ Note that this is a statement applies to all x-values. The inverse fourier transform of $g(x)$ should give $f(p)$, $$ f(p)=\frac{1}{2\pi}\int dx\exp(-ipx)g(x) $$ As $g(x)=0$ according to to $(*)$ then $f(p)=0$.


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Yes, one traditional alternative to the path integral formalism is the operator formalism. For QED with abelian gauge group, the old quantization formulation is the Gupta-Bleuler formulation. For QCD/Yang-Mills theory with non-abelian gauge group, the Gupta-Bleuler formulation is replaced by the BRST formulation. The BRST formulation exists in at least 3 ...


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If you have two terms then you would have two vertices which contribute to a certain graph. For your first term, as far as I can tell, you would have a vertex for $\phi_i+\chi_j\to\phi_k+\chi_l$ given by $\propto g_3^2(\lambda^a)_{ik}(\lambda^a)_{jl}$. The general recipe to derive the Feynman rules is to feed your Lagrangian into the path integral and just ...


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Found a sketch of a proof on a referee's report on a paper RELATIVISTIC INVARIANCE OF THE VACUUM by Adam Bednorz. The referee's sketch is: Comment Hundreds of calculations in Fnite temperature Feld theory have been published. To my knowledge, none of these calculations have ever conflicted with Lorentz invariance in the limit $\beta \to \infty$ ...


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Where a string carves out a $2$-dimensional world-sheet and a point particle carves out a $1$-dimensional world-line of spacetime, the instanton carves out a $0$-dimensional world-point. Counting only spatial dimensions, a string is $1$-dimensional and a point particle is $0$-dimensional. By logical extension, an instanton has dimension $-1$, if we only ...


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A late answer, but important in my opinion. There is another aspect: the sign in the covariant derivative also depends on the sign convention used in the gauge transformation! This is something that is overlooked a lot. If the Dirac field transforms as $$ \psi \rightarrow e^{ig\alpha} \psi, $$ then the covariant derivative is defined as $$ D_\mu = ...


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In $d=4$ and $n=3$, you the following relation $w = 4 - E - V$, where: $w=$ supercifial degree of divergence $E=$ external legs $V=$ number of vertices So, if you replace the values of $d$ and $n$, you will get $w<0$, therefore you won't have $UV$ divergences in that diagram.


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There exists an extensive literature for discretization of the abelian and the non-abelian gauge theories, known as lattice QED and lattice QCD, respectively. Here we will only sketch the main idea. Let us for simplicity use Euclidean signature $(+,+,+,+)$. A small Wilson-loop $$\tag{1} W~=~{\rm Tr}{\cal P}e^{ig\int_{\gamma}A}$$ lies approximately in a ...


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My answer is more of comment on other correct answers: you cannot build a delta-function for the photon in 3D becase the longitudinal component of a massless vector field is missing. But that does not mean there is no useful and meaningful concept of a wave function in the single-photon sector. This is just a peculiar fact about free electromagnetic field, ...


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There are several different waves associated with a photon. In QED the photon is associated with a classical solution of the (4-)vector potential. The vector potential contains features that are not physical, as a change of gauge is not reflected in any change of physical properties. Thus its role as a wave function might be somewhat questionable.But still, ...


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To be honest, we were somewhat surprised when reading your posts. You talk of "gravitomagnetism". This is the theory of Oliver Heaviside which he published in 1893 and it is the Maxwell-analogy for gravity. However, it is known that the general relativity theory is based upon a totally different set of premises than gravitomagnetism. Also, it is known that ...


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Actually, when I was studying quantum mechanics, I was surprised that some everywhere nonzero potentials produce only a finite total cross section. It is intriguing because classically, for such potentials, any particle injected will be scattered, although might with a very small deflection angle. I mean, it is not surprising to have an infinite total ...


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I am very ignorant here, but.. Surely the question is whether you want to do statistical mechanics or not. Finite temperatures will allow you to ignore the energy sources - but still enable you to calculate the production of particles. I don't see how you would ever use it for scattering problems involving small numbers of particles. On Noldig's answer: ...



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