How Should I Think About the Dirac Equation

[bolbteppa]

In Weinberg's QFT Vol. 1 he says the Dirac equation is not a true generalization of Schrodinger's equation, that it does not stand up to inspection when viewed in this light. He says it should be viewed as an approximation to a true relativistic quantum field theory of photons and electrons.

a) I do not understand what this means, would someone mind filling me in?

One of Dirac's motivations for his derivation was that Klein-Gordon was not first order in time like the non-relativistic Schrodinger equation is.

b) Since the 2'nd order in time Klein-Gordon equation does actually describe something physical.does this mean Dirac's point about not being first-order in time is actually flawed? From browsing Cartan's Spinor's book it seems the Dirac equation holds for any spinor, it apparently relates left & right representations of a spinor or something, thus it holds in GR etc... There is also this great quote from Atiyah that a spinor is a square root of a geometry.

c) What is the Dirac equation & how does this explain why Dirac's derivation worked, why it relates representations of a spinor & explains this square root of a geometry business?

 

[stevendaryl]

In Weinberg's QFT Vol. 1 he says the Dirac equation is not a true generalization of Schrodinger's equation, that it does not stand up to inspection when viewed in this light. He says it should be viewed as an approximation to a true relativistic quantum field theory of photons and electrons.

a) I do not understand what this means, would someone mind filling me in?

One of Dirac's motivations for his derivation was that Klein-Gordon was not first order in time like the non-relativistic Schrodinger equation is.

b) Since the 2'nd order in time Klein-Gordon equation does actually describe something physical.does this mean Dirac's point about not being first-order in time is actually flawed?From browsing Cartan's Spinor's book it seems the Dirac equation holds for any spinor, it apparently relates left & right representations of a spinor or something, thus it holds in GR etc... There is also this great quote from Atiyah that a spinor is a square root of a geometry.

c) What is the Dirac equation & how does this explain why Dirac's derivation worked, why it relates representations of a spinor & explains this square root of a geometry business?

I think that what Weinberg was talking about was that it is difficult to interpret the Dirac equation as an equation for a single particle, for several reasons. One is the fact that it has an infinite number of negative energy levels, so there is no "ground state". Dirac fixed this by postulating a "sea" of negative-energy electrons in what we would normally think of as "the vacuum", but such a sea means that you never really have a one-particle system.

 

[bolbteppa]

He could have said the same thing about the Klein-Gordon equation, to say it was not a relativistic generalization of Schrodinger. I think it's more subtle than that.

 

[king vitamin]

In Weinberg's QFT Vol. 1 he says the Dirac equation is not a true generalization of Schrodinger's equation, that it does not stand up to inspection when viewed in this light. He says it should be viewed as an approximation to a true relativistic quantum field theory of photons and electrons.

a) I do not understand what this means, would someone mind filling me in?

The problem is that Dirac applied his equation to a single particle, while a relativistic theory should really account for the creation and destruction of particles. Sidney Coleman makes a similar point in his first QFT lecture (videos here: https://www.physics.harvard.edu/events/videos/Phys253 ); since the formalism doesn't allow multi particle electron/photon states, it's missing possible effects which effect the energy levels. IIRC, Coleman call's Dirac's result a "fluke," claiming it only works because certain QED effects are small.

b) Since the 2'nd order in time Klein-Gordon equation does actually describe something physical.does this mean Dirac's point about not being first-order in time is actually flawed?

Weinberg seems to think so. I think Dirac's work is a brilliant bit of physics, finding the correct results by inspired guesswork which is arguably incorrect in retrospect. It turns out that the Dirac and Klein Gordon equations are important for relativistic fields (operators which are functions of spacetime), and relativistic states evolve under the (first-order in time) Schrödinger equation with a Hamiltonian which is a Lorentz scalar made up of these fields.

From browsing Cartan's Spinor's book it seems the Dirac equation holds for any spinor, it apparently relates left & right representations of a spinor or something, thus it holds in GR etc... There is also this great quote from Atiyah that a spinor is a square root of a geometry.

c) What is the Dirac equation & how does this explain why Dirac's derivation worked, why it relates representations of a spinor & explains this square root of a geometry business?

Well, Weinberg's book answers all of this, but it's a long road since Weinberg insists on rigor. The last chapter of Weinberg Vol. 1 derives the Dirac equation from quantum field theory (using the Bethe-Salpeter equation if i remember correctly), using a method which allows full QED corrections (he computes the Lamb shift).

The fully relativistic viewpoint is what I alluded to above. The state evolves under the standard Schrödinger equation, id|ψ>/dt = H|ψ>, but for this to make sense relativistically, the form of the Hamiltonian is massively constrained, and you need to introduce relativistic quantum field theory. The fields required to create spin-1/2 states must transform in spinor representations of the Lorentz group, and the free part of the Hamiltonian involves the Dirac equation in an essential way (for spin-0 particles, the fields must be scalars, and the free Hamiltonian involves Klein-Gordon). QFT develops perturbation theory using Feynman diagrams, and in order to compute bound states (like the Hydrogen atom), you must sum an infinite set of these diagrams (since you can't get a bound state from a finite order in perturbation theory). This method is the Bethe-Salpeter equation, and for a Coulomb potential, you can show that the bound state energies are computed from solving the Dirac equation. But the philosophy is totally different from the original Dirac paper.

 

[bolbteppa]

Weinberg seems to think so. I think Dirac's work is a brilliant bit of physics, finding the correct results by inspired guesswork which is arguably incorrect in retrospect. It turns out that the Dirac and Klein Gordon equations are important for relativistic fields (operators which are functions of spacetime), and relativistic states evolve under the (first-order in time) Schrödinger equation with a Hamiltonian which is a Lorentz scalar made up of these fields.
...

The fully relativistic viewpoint is what I alluded to above. The state evolves under the standard Schrödinger equation, id|ψ>/dt = H|ψ>, but for this to make sense relativistically, the form of the Hamiltonian is massively constrained, and you need to introduce relativistic quantum field theory. The fields required to create spin-1/2 states must transform in spinor representations of the Lorentz group, and the free part of the Hamiltonian involves the Dirac equation in an essential way (for spin-0 particles, the fields must be scalars, and the free Hamiltonian involves Klein-Gordon).

I thought the point of QFT was to derive Klein-Gordon, Dirac & Proca equations as relativistic generalizations of Schrodinger then second quantize them to express these equations in terms of fields (or do it all with path integrals) - nowhere have I seen the non-relativistic Schrodinger equation! Dyson's notes seem to say it was disbanded because it singles out time on the l.h.s. in id|ψ>/dt = H|ψ> making Lorentz invariance extremely difficult, thus K-G, Dirac & Proca are used instead. What am I missing?

 

[vanhees71]

Weinberg is right in saying that the Dirac equation cannot be interpreted in terms of a relativistic single-particle quantum theory in analogy to the Schrödinger or Pauli equations of non-relativistic physics. As far as I know, there is no working example of such a relativistic single-particle theory. Relativistic quantum theory leads to the notion that each particle has its antiparticle "counter part" (with the exception of strictly neutral particles which are identical with their antiparticles), if you make the assumption that a stable ground state exists (the Hamiltonian should be bounded from below).

The single-particle interpretation of the Dirac equations becomes a good approximation in the non-relativistic limit. There you get relativistic corrections to the corresponding non-relativistic approximation in terms of a Schrödinger- or Pauli equation. E.g., the atomic bound state problem (e.g., for hydrogen, where the binding energies are small compared to the mass of the electron) can be well described with an electron moving in the (static Coulomb) field of the atomic nucleus. In addition you can evaluate QED corrections starting from this approximation, leading to results that are among the best established results with the best agreement between theory and experiment in physics ever.

Historically Dirac came to the correct interpretation of relativistic QED through his equation in terms of the "hole theory". This theory turns out to be equivalent to modern QED based on the usual perturbative techniques of QFT, but it's pretty complicated compared to the more elegant QFT formulation. Weinberg seems to have some aversion against Dirac. You can read pretty funny statements against Dirac in the prefaces of his QFT textbook (vol. 1) and his newer textbook on non-relativistic quantum theory. There is, however, no doubt that Dirac is among the best theoretical physicists of all times and among the pioneers of quantum theory (both non-relativistic and relativistic) the one with the most clear exihitions of the theory already in his original research papers. His socalled "transformation theory", which is more or less nothing else than the modern representation-independent formulation (the mathematically stricter foundation in terms of Hilbert-space theory is due to von Neumann).

 

[atyy]

I thought the point of QFT was to derive Klein-Gordon, Dirac & Proca equations as relativistic generalizations of Schrodinger then second quantize them to express these equations in terms of fields (or do it all with path integrals) - nowhere have I seen the non-relativistic Schrodinger equation! Dyson's notes seem to say it was disbanded because it singles out time on the l.h.s. in id|ψ>/dt = H|ψ> making Lorentz invariance extremely difficult, thus K-G, Dirac & Proca are used instead. What am I missing?

That's the old confused sense of second quantization. Nowadays the term "second quantization" is still used, but the conception is different. In principle, there is no quantization, since the quantum theory comes first, and the classical theory is the limit. However, while we are guessing quantum theories from classical ones, a very good way is Dirac's method of quantization. Then first quantization is the Dirac quantization of a classical system of particles, and second quantization is the Dirac quantization of a classical system of fields. This makes complete sense when we second quantize the Maxwell photon field. It is still ok for the Dirac electron field, except that it's appears more of a trick here since there is no classical electron field. Another way to see that there is no classical electron field is that in path integral quantization, for the electromagnetic field one can use classical variables, but for fermions one has to use Grassmann variables, even though the path integral is a (sometimes rigourous) trick that makes quantum mechanics look classical.

The second sense of second quantization that is used nowadays is that it is a way of exactly rewriting the non-relativistic Schroedinger equation for many identical particles as a quantum field theory. Because it is possible to rewrite non-relativistic quantum mechanics in this way, the non-relativistic theory may be "derived" as an approximation to relativistic quantum field theory in the regime in which particles are neither created nor destroyed, the particles can be considered to be in an external potential, and Galilean symmetry holds to a very good approximation. I don't think we can do a proper derivation, so mainly we know such a regime exists, and then we write all possible terms consistent with the Galilean symmetry, and use as few as we need to fit the experimental data.

 

[atyy]

Here is the second quantized Schroedinger equation. http://hitoshi.berkeley.edu/221b/QFT.pdf

By treating the Schroedinger equation as a classical field, we obtain the quantum mechanics of many identical particles. The difference between non-relativistic quantum field theory obtained by second quantizing the Schroedinger field, and a relativistic quantum field theory obtained by second quantizing the Klein-Gordon, Maxwell or Dirac fields is that the number of particles is fixed in the non-relativistic theory.

So non-relativistic quantum mechanics of many identical particles can be obtained in two ways.
(1) First quantize a classical system of identical particles
(2) Second quantize a classical Schroedinger field

It may seem that the second quantized language is more fundamental since relativistic field theories are more fundamental. However, the standard model is not UV complete, and many aspects of it can be modeled by a lattice theory, which does have a first quantized description. The big problem for a lattice description is chiral interactions.

 

[bolbteppa]

Okay I have to be more careful. There is a non-relativistic second quantization of the many particle Schrodinger equation (Landau Vol. 3, sec 64 & 65). Then there is a relativistic second quantization which involves deriving Klein-Gordon, Dirac & Maxwell-Proca for particles then turning them into fields & invoking commutation relations (ala Peskin & Schroeder), something you can also do with path integrals - I assume Dirac's little book on quantization is just following this method (without path integrals), is he? It looks different to Peskin & Schroeder anyway...

Dyson's notes on QM (pdf page 40) mention some old method of deriving a relativistic theory, which I think is just plugging in a relativistic Hamiltonian into the non-relativistic Schrodinger equation id|ψ>/dt = H|ψ> which he says is bad. I've never seen a QFT book use the non-relativistic Schrodinger equation, king vitamin's comments make it seem like QFT is ultimately just ruled by the non-relativistic Schrodinger id|ψ>/dt = H|ψ> and that all the second quantization stuff is just something extra to account for relativity. What is the non-relativistic Schrodinger equation doing in QFT?

 

[atyy]

The general Schroedinger equation id|ψ>/dt = H|ψ> applies in quantum field theory also. However, the state has to be considered in Fock space, and the operators in the Hamiltonian are operators on Fock space. However, time evolution in the Schroedinger picture is usually hard to calculate with, so one uses either the Heisenberg picture or the interaction picture for calculations. As in non-relativistic mechanics, the predictions of all these different pictures are the same. (If the Schroedinger picture is used in field theory, people usually use the Schroedinger functional language.)

(There's a famous problem (Haag's theorem) with the standard arguments in truly rigourous relativistic quantum field theory, but it turns out the wrong derivations give the right equations anyway. But in practice, if one assumes that our field theories are not truly rigrourous and relativistic, and just represent a fine lattice that occupies a huge but finite space, then everything is just an efficient way of calculating for the quantum mechanics of many identical particles.)

 

[dextercioby]

For atyy's comment in post#10, check out chapter 10 of B. Hatfield's <The Quantum Field Theory of Point Particles and Fields>.

 

[king vitamin]

king vitamin's comments make it seem like QFT is ultimately just ruled by the non-relativistic Schrodinger id|ψ>/dt = H|ψ> and that all the second quantization stuff is just something extra to account for relativity. What is the non-relativistic Schrodinger equation doing in QFT?

Maybe I should've been more careful. In general, even in non-relativistic QM, you don't need your states to evolve under Schrödinger's equation id|ψ>/dt = H|ψ>. Generally, you need your observables to evolve under time, and there are a few different ways to do this: have your states evolve under Schrödinger's equation, have your operators evolve under Heisenberg's equation, or you have some mixture in the interaction pictures (operators evolve under Heisenberg's equation, states evolve under a Dyson series evolution operator). This is all still exact in relativity, but since the major constraint you're trying to enforce is causality, you usually look in the Heisenberg representation: make every operator a function of spacetime which commutes outside of each other's light cone. (In fact, this constraint is required to be consistent with unitary time evolution, see Weinberg Chapter 3).

In perturbation theory, you usually go to the interaction picture, where id|ψ>/dt = H_int|ψ> and H_int is the interacting part of the Hamiltonian. You claim that this seems absent from QFT, but you also say you have Peskin & Schroeder - just multiply equation 4.18 in P&S by your state to see that it satisfies this Schrödinger equation. The form of this equation is simply a consequence of evolution under a unitary operator. I also didn't see where Dyson contradicts this on page 40 of the notes you linked - indeed, he seems to be very clear about it's importance (equations 241, 245 and 417 for instance).

(I ignore Haag's theorem everywhere)

 

[dextercioby]

A bottom line might be that id|ψ>/dt = H|ψ> doesn't mean automatically Galilean relativity (the so-called 'non-relativistic physics').
If H is a self-adjoint operator, it simply means that the Hamiltonian is the generator of dynamics, or of time-evolution of quantum states, in other words it's the generator of a 1-parameter strongly continuous subgroup of the symmetry group of the space-time which can be: Galilei for classical mechanics, Poincare for specially relativistic quantum (field) theories, the conformal group for the conformal field theories.

 

[bolbteppa]

It all makes a little bit more sense now, still need to think it all over for a while - thanks for the Hatfield recommendation it's actually amazing.