Three weeks ago, inward the article titled
Responsibilityphysicist Jacques Distler of UT Austin mentioned a disceptation past times Sasha Polyakov that he was "responsible" for quantum land theory. That comment was specially relevant when Distler taught an undergraduate particle physics course of pedagogy in addition to was frustrated past times the following:
The textbooks (and I hateful all of them) start off past times “explaining” that relativistic quantum mechanics (e.g. replacing the Schrödinger equation amongst Klein-Gordon) brand no feel (negative probabilities in addition to all that …). And they so continue to utilization it anyway (supplemented past times to a greater extent than or less Feynman rules pulled out of sparse air).Did the next text defend the legitimacy of Distler's frustration? Well, partly... but I would alternative the respond No if I had to.
This drives me upward the fúçkïñg wall. It is exactly wrong.
There is a perfectly consistent quantum mechanical theory of gratis particles. The occupation arises when you lot want to innovate interactions.
What's going on? Indeed, textbooks in addition to instructors oftentimes – and, according to to a greater extent than or less measures, e'er – say that quantum mechanics of 1 particle ceases to lead good 1 time you lot switch to relativity – to theories covariant nether the Lorentz transformations.
Are these statements right? Are they wrong? And are the right statements 1 tin sack brand important? It depends what exact statements you lot accept inward mind.
What Distler discusses is the existence of the Hilbert infinite – in addition to Hamiltonian – for 1 particle, e.g. the Klein-Gordon particle. Does it exist? You bet. If you lot believe that a Hilbert infinite of particles exists inward gratis quantum land theory, practise the following: Write a solid soil vector of that Hilbert infinite as the solid soil of a Fock space, i.e. inward damage of the solid soil vector that are\[
a^\dagger_{\vec k_1} \cdots a^\dagger_{\vec k_n} \ket 0
\] And merely alternative those solid soil vectors that comprise exactly 1 creation operator. This one-particle subspace of the Hilbert infinite volition evolve to itself nether the empty-spacetime development operators. In fact, if you lot write the solid soil inward the momentum solid soil as I did, the Hamiltonian for 1 existent quantum of the existent Klein-Gordon equation volition survive simply\[
H = \sqrt{|\vec k|^2 + m^2}.
\] This is something you lot may derive from quantum land theory. The operator higher upward is perfectly well-defined inward the momentum space. The unloosen energy is non-negative, the norms of states are positive, everything plant fine.
So has Distler shown that all the statements of the type "one particle isn't consistent inward relativistic quantum mechanics" are wrong?
Nope, he hasn't. In particular, he was talking most the statement
...replacing the [non-relativistic, e.g. one-particle] Schrödinger equation amongst Klein-Gordon make[s] no sense...But this disceptation is right at the score of one-particle quantum mechanics because his equation for the development of the moving ridge business office is not the Klein-Gordon equation. You know, the Klein-Gordon equation is\[
\left(\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} + m^2 \right) \Phi = 0.
\] That's a nice, local – perfectly differential equation. On the other hand, the replacement for the non-relativistic Schrödinger equation\[
i\hbar\frac{\partial}{\partial t} \psi = -\frac{\hbar^2}{2m} \Delta \psi + V(x) \psi
\] that he derived in addition to that describes the development of one-particle states was\[
i\hbar\frac{\partial}{\partial t} \psi = c \sqrt{m^2c^2-\hbar^2\Delta} \psi + V(x) \psi
\] Because the foursquare root has a neverending Taylor expansion, the business office of the Laplace operator is a terribly non-local "integral operator" acting on the moving ridge business office \(\psi(x,y,z,t)\) inward the seat representation. So this equation for 1 particle, fifty-fifty though it follows from the Klein-Gordon quantum land theory, doesn't accept the overnice in addition to local Klein-Gordon form. It isn't pretty in addition to it isn't fundamental. If you lot wrote this equation inward isolation, you lot should survive worried that the resulting theory isn't relativistic because relativity implies locality in addition to this equation allows the localized moving ridge business office bundle to spread superluminally!
What the statements hateful is that if you lot want to utilization to a greater extent than or less overnice in addition to local equation for a moving ridge business office for 1 particle – i.e. if you lot literally want to supplant Schrödinger's equation past times the similar Klein-Gordon equation – you lot won't uncovering a manner to build (in damage of local functions of derivatives etc.) the probability electrical flow in addition to density etc. that would accept the desired positivity properties etc. And this disceptation is just truthful in addition to important!
If you lot want to provide to simple, fundamental, justifiable, beautiful equations, you lot tin sack indeed utilization the Klein-Gordon, Dirac, Maxwell, in addition to other equations. But you lot must appreciate that they're equations for (field) operators, non for moving ridge functions.
This disceptation is of import because it's non just a mathematical one. It's highly physical, too. In particular, if you lot consider whatever relativistic quantum mechanical theory of particles – quantum land theory or something grander, similar string theory – it's unavoidable that when you lot restrain particles to the distance shorter than the Compton wavelength \(\hbar / mc\) of that particle, you lot volition unavoidably accept plenty unloosen energy so that particle-antiparticle pairs volition start to survive produced at nonzero probabilities. And inward relativity, it's normal for a item to motion past times a speed comparable to the speed of light, in addition to so its wavelength is comparable to the Compton wavelength. You can't really trust the one-particle theory at distances comparable to its normal de Broglie wavelength! So the theory is incorrect inward to a greater extent than or less really rigid sense.
The antiparticles (which are the same amongst the master copy particle inward the existent Klein-Gordon case, just to survive sure) inevitably follow from relativity combined amongst quantum mechanics, in addition to so does the span production of particles in addition to antiparticles. This physical disceptation has lots of nearly equivalent mathematical manifestations. For example, local observables inward a relativistic quantum theory have to be constructed out of quantum fields. So the 1-particle Hilbert infinite doesn't accept whatever genuinely local observables: You can't build the Klein-Gordon land \(\Phi(x,y,z,t)\) out of operators acting on the 1-particle Hilbert infinite because the latter operators never alter the pose out of particles piece \(\Phi(x,y,z,t)\) does (by 1 or minus 1 – it's a combination of creation in addition to annihilation operators). In fact, you lot can't build the bilinears inward \(\Phi\) and/or its derivatives, either, because piece those operators inward QFT comprise to a greater extent than or less damage that save the pose out of particles, they also comprise as of import damage that alter the pose out of particles past times ii (particle-antiparticle span production or span annihilation) in addition to those are as of import for obtaining the right commutators in addition to other things. The mixing of creation operators for particles in addition to the annihilation operators for antiparticles is absolutely unavoidable if you lot want to define observables at points (or regions smaller than the Compton wavelength).
There's 1 to a greater extent than disceptation that Distler made in addition to that is really wrong. Distler wrote that the problems alone laid about when you lot start to consider interactions – in addition to from the context, it's clear that he meant interactions involving several quanta of quantum fields, several particles inward the quantum land theory sense. But that's non true.
Problems of "one-particle relativistic quantum mechanics" already look if you lot consider the behaviour of the unmarried particle inward external classical fields. Just crush a Klein-Gordon particle – e.g. a Higgs boson – inward betwixt ii metallic element plates whose distance is sent to zero. Will it brand sense? No, as I mentioned, the walls start to create particle-antiparticle quanta inward general. Time-dependent Hamiltonians Pb to particle production, if you lot wish. Similarly, if you lot house these particles inward whatever external classical field, the actual Klein-Gordon land may react inward a manner to create particle pairs.
So the truncation of the Hilbert infinite of a quantum land theory to the one-particle subspace is inconsistent non alone if you lot consider interactions of particles inward the commons Feynman diagrammatic feel – but fifty-fifty if you lot consider the behaviour of the particle inward external classical fields. Whatever you lot endeavor to amongst the particle that goes beyond the stupid elementary unmarried free-particle Hamiltonian volition forcefulness you lot to admit that the truncated one-particle theory is no good.
We want to practise something to a greater extent than amongst the theory than just write an unmotivated non-local Hamiltonian of the form \(H\sim \sqrt{m^2+p^2}\) if I utilization \(\hbar=c=1\) units here. And as presently as nosotros practise anything else – justify this ugly in addition to seemingly non-local (and thus seemingly relativity-violating) Hamiltonian past times an elegant theory, written report particle interactions, written report the behaviour of 1 particle inward external classical fields – nosotros just demand to switch to the full-blown quantum land theory, otherwise our musings volition survive inconsistent.
One extra comment. I mentioned that the non-local differential operator allows the moving ridge bundle to spread superluminally. How is it possible that such a affair results from a relativistic theory? Well, quantum land theory has no occupation amongst that because when you lot practise whatever doable measurement, the processes inward which a particle spreads inward the optic gets combined amongst processes involving antiparticles. When you lot calculate the "strength of influences spreading superluminally", to a greater extent than or less Green's functions – which are nonzero for spacelike separations – volition combine to the "commutator correlation function" which vanishes at spacelike separation. So the inseparable presence of antiparticles volition salve the locality for you. The truncation to particles-only (without antiparticles) would indeed violate locality required past times relativity as long as you lot could experimentally verify it (you demand at to the lowest degree to a greater extent than or less interactions of that particle amongst something else for that).
While Jacques is right most the possibility to truncate the Hilbert infinite of quantum land theories to the one-particle subspaces, he's morally incorrect most all these big statements – in addition to to a greater extent than or less of his statements are literally wrong, too. At to the lowest degree morally, the lore that drives him upward the wall is right in addition to at that topographic point are ways to formulate this lore so that it is both literally truthful in addition to important, too.
So students inward Austin are encouraged to actively ignore their grumpy instructor's tirades against the quantum land theory lore in addition to fifty-fifty to a greater extent than encouraged to empathise inward what feel the lore is true.
As I explicate inward the comments, many quantum land theory textbooks accept wonderful explanations – unremarkably at the really showtime – of the wisdom that Jacques Distler seems to misunderstand, namely why quantum fields in addition to the mixing of sectors amongst dissimilar numbers of particles is unavoidable for consistency of quantum mechanics amongst special relativity.
The 2008 textbook past times my adviser Tom Banks starts the explanation on Page 3, inward department "Why quantum land theory?" It says that the probability aAmplitude for a particle emission at spacetime signal \(x\) in addition to its absorption at signal \(y\) is unavoidably nonzero for spacelike separations in addition to because it would alone survive alone nonzero for 1 of the ii fourth dimension orderings of \(x,y\), in addition to the ordering of spacelike-separated lawsuit isn't Lorentz-invariant, the Lorentz invariance would survive broken in addition to 1 must really demand that alone amplitudes where both orders are summed over are allowed. In other words, as argued on page 5, the alone known consistent ways to solve this clash amongst the Lorentz invariance is to demand that every emission source must also survive able to deed as an absorption sink in addition to vice versa. When both damage are combined, the marrow is yet nonzero inward the spacelike share but has no vicious discontinuities when the ordering gets reversed.
Also, when the particle carries charges, the emission in addition to absorption inward the ii related processes must involve particles of opposite charges in addition to 1 predicts (and Dirac predicted) the being of antiparticles that are needed for things to work.
Weinberg QFT Volume 1 explains the negative probabilities in addition to energies of the relativistic equations naively used instead of the non-relativistic Schrödinger equation on pages 7, 12, 15... Read it for a while. It's OK but, inward my opinion, much less deep than Tom's presentation.
Peskin's in addition to Schroeder's textbook on quantum land theory discusses the non-vanishing of the amplitudes inward the spacelike share on page xiv in addition to pages 27-28 hash out that the actual influence of 1 criterion on to a greater extent than or less other is measured past times the commutator of ii land operators. And that vanishes for spacelike separations – again, because ii processes that are opposite to each other are subtracted.
Without the mixing of creation operators (for particles) in addition to annihilation operators (for antiparticles), you lot just can't define whatever observables that would belong to a signal or a share in addition to that would lead relativistically (respected the independence of observables that are spacelike separated). Quantum fields are the alone known manner to avoid this conflict betwixt quantum mechanics in addition to relativity. They are unavoidably superpositions of positive- in addition to negative-energy solutions, in addition to thus are expanded inward sums of creation in addition to annihilation operators. That's why all local discussions come inward necessary to let emission in addition to absorption at the same fourth dimension – and, consequently, the combination of quantum mechanics in addition to relativity makes it necessary to consider the whole Fock infinite amongst a variable pose out of particles. The one-particle truncation is inconsistent amongst relativistic dynamics such as time-dependent interactions, emission, or absorption.
In the mathematical language, fields in addition to their functions are necessary for whatever local observables inward relativistic quantum mechanical theories. They e'er comprise damage that alter the pose out of particles – except for the footling constant operator \(1\). In the physical language, relativity in addition to quantum mechanics simultaneously imply that emission in addition to absorption are linked, antiparticle exists, in addition to scattering amplitudes for particles in addition to antiparticles accept to obey identities such as the crossing symmetry.
The teaching of a quantum land theory course of pedagogy could survive a expert chance for Jacques to larn this basic materials that is oftentimes presented on pages such as 3,5,7,12,14... of introductory textbooks.
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