Deriving the Dirac equation from an action principle

[Ben Niehoff]

Some confusions from some recent lectures; I asked the prof, but I still don't fully understand what is going on. We began with the action (tau is some worldline parameter, dots indicate tau derivatives; they are hard to see):

$$S = \int d\tau \; \left\{ \dot x^{\mu} p_{\mu} - \frac12 e(\tau) (p^2 - m^2) + f(\tau) (p_{\mu} \lambda^{\mu} - m) + i \dot \lambda_{\mu} \lambda^{\mu} \right\}$$

Varying this action with respect to the five dynamical variables gives the equations of motion

$$\dot p_{\mu} = 0$$

$$\dot x^{\mu} - e(\tau) p^{\mu} + f(\tau) \lambda^{\mu} = 0$$

$$p^2 - m^2 = 0$$

$$f(\tau) p_{\mu} = 0$$

$$p_{\mu} \lambda^{\mu} - m = 0$$

Promoting x and p to operators and imposing the standard commutation relations, the first three equations (with f = 0) give the Klein-Gordon equation. This much I understand.

However, to arrive at the Dirac equation, the prof's next step is simply to impose anti-commutation relations on $\lambda^{\mu}$, without any motivation, as far as I can tell. Can anyone clarify the reason for doing this? It seems completely arbitrary.

Also, where do the last two pieces in the Lagrangian come from? The first term serves to link x and p as canonical conjugates, and the second term is a restatement of the mass-shell constraint. But the last two terms don't seem to have any prior justification at all. Are they just thrown in there because they result in the Dirac equation? If so, the entire "derivation" is just a circular argument. If the last two terms are put in because of some more fundamental principle, what would that be?

 

[per.sundqvist]

I can't read your equation, but it seems its missing a square root. From my knowledge the relativistic Lagrangian is given by (1D for simplicity):

$$L=m_0c^2\sqrt{1-\left(\frac{\dot{x}(t)}{c}\right)^2}+q\vec{A}\cdot\dot{\vec{x}}-q\Phi(x)$$

I remember I learned to derive the Schrödinger equation from Feynman path integral formulation from the kernel of action $$\exp(i\;S[a,b]/\hbar)$$. It gives the correct answer to lowest order in space derivative (laplacian). Here we must in some way also reflect over that this kernel will act on a four element spinor, which is a bit ad hoc, but maby it is possible?

 

[Peeter]

I can't read your equation, but it seems its missing a square root.

The root is not needed if the Lagrangian is given if the derivatives are taken with respect to proper time.

One place to find an E&M example of such a covariant Lagrangian (simpler than the Dirac equation) can be found here:

http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf

 

[per.sundqvist]

The root is not needed if the Lagrangian is given if the derivatives are taken with respect to proper time.

One place to find an E&M example of such a covariant Lagrangian (simpler than the Dirac equation) can be found here:

http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf

That was interesting to read, but it sounds almost to good to be true...? The Hamiltonian was also like mv^2/2, so my question is: Why didn't Dirac use this instead? There must be some problem with this, with some extra condition, or?

 

[Ben Niehoff]

That is interesting, but I think it's a bit of a tangent. I'm talking about a free field theory here, without electromagnetic interactions. The Lagrangian is parametrized along the worldline. I know it's not the usual Lagrangian; that's why I was asking about it. Apparently it has something to do with supersymmetry, but the prof did not really go into that at all.

I think my real question boils down to: Why do we use anticommutators to quantize the Dirac field? I mean, yes, I know that anticommutation relations will give us fermions, but as far as I can tell, we are just throwing them in arbitrarily. Is there any sense in which the theory naturally leads to fermions, without putting them in by hand?

Apparently this has something to do with the Spin Statistics Theorem, but I don't really understand how it works. If anybody has any clues, that would be nice.

 

[Bob_for_short]

I think that the least action principle is not fundamental but secondary. The Lagrangian or Hamiltonian actions were constructed after establishing the differential (Newton) equations of motion. As well, any physical problem includes the initial conditions. In the action there is no place for the initial velocity; it is the final position which is given. This does not correspond to physical situation. Yes, for many differential equations in physics the corresponding actions can be constructed. The symmetries and the conservation laws are also much easier obtained from the action principle. But let us not forget that most of equations in physics were first formulated without action principle - as the experimental data fitting. Unfortunately mathematicians put this upside down: now the action principle is the primary (fundamental), the equations are secondary (derived). We have a lot of problems with such an approach (see my popular article on it at arxiv:0811.4416, "Reformulation instead of Renormalizations" by Vladimir Kalitvianski).

Concerning anticommutators, on the quantum level we must employ the experimental facts: all elementary particles are bosons or fermions, especially if the statistics plays a significant role. This, in addition to the differential equations and the initial data, completes the physical picture; no wonder it is required in the mathematical apparatus.

Vladimir.

 

[Eredir]

I think my real question boils down to: Why do we use anticommutators to quantize the Dirac field? I mean, yes, I know that anticommutation relations will give us fermions, but as far as I can tell, we are just throwing them in arbitrarily. Is there any sense in which the theory naturally leads to fermions, without putting them in by hand?

Apparently this has something to do with the Spin Statistics Theorem, but I don't really understand how it works. If anybody has any clues, that would be nice.

A valid reason could be that we need to impose anti-commutation relations for the Hamiltonian to be bounded below, which is a reasonable physical requirement.

From the normal mode expansion
$$\psi(x) = \sum_{p,r}\sqrt{\frac{m}{E(p)V}}[a_{r}(p)u_{r}(p)e^{-ipx} + b_{r}^{\dagger}(p)v_{r}(p)e^{ipx}]$$
one can obtain the Hamiltonian
$$H = \sum_{p,r} E(p) [a_{r}^{\dagger}(p) a_{r}(p) - b_{r}(p) b_{r}^{\dagger}(p) ]$$.

Using anti-commutation relations we rewrite this (apart from the usual constant) as
$$H = \sum_{p,r} E(p) [a_{r}^{\dagger}(p) a_{r}(p) + b_{r}^{\dagger}(p) b_{r}(p) ]$$
and this Hamiltonian is bounded below. If instead we used commutation relations this wouldn't be true.

 

[Ben Niehoff]

Concerning anticommutators, on the quantum level we must employ the experimental facts: all elementary particles are bosons or fermions, especially if the statistics plays a significant role. This, in addition to the differential equations and the initial data, completes the physical picture; no wonder it is required in the mathematical apparatus.

A valid reason could be that we need to impose anti-commutation relations for the Hamiltonian to be bounded below, which is a reasonable physical requirement.

... If instead we used commutation relations this wouldn't be true.

Yes, I see, but these are physical arguments rather than mathematical ones. In the interest of reducing a theory to the smallest number of assumptions possible, shouldn't we hope to devise some simple principle out of which the spin-statistics correspondence arises by necessity?

Similar to what Bob (Vladimir) says, using anticommutators "because they give a Hamiltonian bounded from below" is to fit assumptions to final conclusions, rather than work them the right way around. Is there any uniqueness theorem to say that anticommutators are the only way to quantize the Dirac field giving well-defined lowest energy state?

Again, I can see that "it works". That's great. But shouldn't we be able to find something more fundamental, rather than just putting in the anticommutation relations by hand?

 

[strangerep]

[...]Yes, I see, but these are physical arguments rather than mathematical ones. In the interest of reducing a theory to the smallest number of assumptions possible, shouldn't we hope to devise some simple principle out of which the spin-statistics correspondence arises by necessity?

Similar to what Bob (Vladimir) says, using anticommutators "because they give a Hamiltonian bounded from below" is to fit assumptions to final conclusions, rather than work them the right way around. Is there any uniqueness theorem to say that anticommutators are the only way to quantize the Dirac field giving well-defined lowest energy state?

Again, I can see that "it works". That's great. But shouldn't we be able to find something more fundamental, rather than just putting in the anticommutation relations by hand?

Well, here's another way of looking at it...

The types of elementary particles can be classified according to the positive-energy
unitary irreducible representations (unireps) of the Poincare group. This encapsulates
the "fundamental" principle of combining special relativity and quantum mechanics.

Proceeding rigorously, one finds that +ve energy half-integer spin unireps (with
infinite degrees of freedom, ie a quantum field) don't exist if you use normal QM
canonical commutation relations for the basic fields, yet do exist if you postulate
canonical anticommutation relations instead.

That about as "fundamental" as it gets with current mainstream theory.
Weinberg vol-I goes through some of the constructions in
his early chapters. More intense rigor can be found in Streater & Wightman.
(Warning: neither is an easy read for anyone just starting into this stuff.)

 

[Ben Niehoff]

Proceeding rigorously, one finds that +ve energy half-integer spin unireps (with
infinite degrees of freedom, ie a quantum field) don't exist if you use normal QM
canonical commutation relations for the basic fields, yet do exist if you postulate
canonical anticommutation relations instead.

Yes, but why not postulate something else entirely? That's the real question, I think.

 

[strangerep]

Yes, but why not postulate something else entirely?

Like what?

 

[Ben Niehoff]

Well, for example, try to find some set of variables on which one can impose ordinary commutation relations and get a consistent, quantized theory.

Or do something different, like

$$AB + iBA = i\hbar$$

 

[strangerep]

Well, for example, try to find some set of variables on which one can impose ordinary commutation relations and get a consistent, quantized theory.

It must also be a theory which has the correct classical limit, and accounts for all the
known relevant experimental phenomena. With ordinary commutation relations, one has
a difficult job to make one's theory produce the Pauli exclusion principle.

Or do something different, like
$$AB + iBA = i\hbar$$

That sort of thing comes under the heading of "quantum deformations" (i.e., deforming
the commutation relations somehow). Some researchers try to find more satisfactory
versions of interacting quantum field theory via that route, so far without conclusive
success.

Different CCRs also arise when a constraint is present (look up "Dirac bracket"
on Wiki for more on that), but one must still impose anticommutation by hand
somewhere for fermions.