What do the psi_3 and psi_4 components of the Dirac equation represent?

[snoopies622]

TL;DR Summary

Seeking a basic understanding of how non-zero values of these components manifest themselves in experiments.

Forgive me if you've heard this song before, but I don't understand how to interpret the $\psi_3$ and $\psi_4$ components of the Dirac equation. For instance, at 8:27 of this video



we see that while an electron at rest can be in a state like [1,0,0,0], the same electron as viewed from a reference frame moving along the x-axis would be seen in the state [1, 0, 0, $p_x / (E+m)$].

Does this mean that in the moving reference frame, there is a non-zero probability of observing a positron instead of (or in addition to) an electron? Somehow the $\psi_3$ and $\psi_4$ components are associated with both positrons and negative energy, which itself confuses me since positrons have positive energy.

I'm also wondering how the Dirac equation is normalized (if that's the right word). When using the Schrodinger equation for a particle, at any given moment $\int |{\psi}|^2 =1$ for all of space because it is certain that the particle will be found somewhere. What is the parallel relation for the Dirac equation? Is it $\int (|{\psi_1}| ^2 + |{\psi_2}|^2 + |{\psi_3}|^2 + |{\psi_4}|^2) =1$ for all of spacetime?

Thanks.

 

[anuttarasammyak]

Normally the four components are {positron, electron} X {spin1/2 up, spin1/2 down}.
So I suppose what you asking is positron X {spin up, spin down}.

 

[PeroK]

Summary:: Seeking a basic understanding of how non-zero values of these components manifest themselves in experiments.

Forgive me if you've heard this song before, but I don't understand how to interpret the $\psi_3$ and $\psi_4$ components of the Dirac equation. For instance, at 8:27 of this video



we see that while an electron at rest can be in a state like [1,0,0,0], the same electron as viewed from a reference frame moving along the x-axis would be seen in the state [1, 0, 0, $p_x / (E+m)$].

Does this mean that in the moving reference frame, there is a non-zero probability of observing a positron instead of (or in addition to) an electron? Somehow the $\psi_3$ and $\psi_4$ components are associated with both positrons and negative energy, which itself confuses me since positrons have positive energy.

You're confusing the third and fourth components with "positron". That is simply not the case. The full relativistic spin state for an electron must have at least these two components - i.e. spin is now a four-dimensional state vector, rather then the two of non-relativistic QM.

A positron would have the "negative energy" solution of the spatial wave-function.

One manifestation of the four-dimensionality of the spinor is that in the relativistic case only the total angular momentum of the electron (orbital plus spin) is conserved; unlike the non-relativistic case where orbital AM and spin AM are separately conserved.

 

[PeroK]

I'm also wondering how the Dirac equation is normalized (if that's the right word). When using the Schrodinger equation for a particle, at any given moment $\int |{\psi}|^2 =1$ for all of space because it is certain that the particle will be found somewhere. What is the parallel relation for the Dirac equation? Is it $\int (|{\psi_1}| ^2 + |{\psi_2}|^2 + |{\psi_3}|^2 + |{\psi_4}|^2) =1$ for all of spacetime?

The spinor components represent the spin state. In principle, therefore, we ought to have a spinor of unit norm. However, it's the entire wave-function (spatial plus spin) that needs to be normalised. There are, therefore, different normalisation conventions for the spinors - having the spinors of unit norm is not the only option. Your course should cover this at some point.

 

[vanhees71]

First of all the question in the title of this thread cannot be answered without quoting the conventions used for the $\gamma$ matrices.

The general answer is that an otherwise unrestricted Dirac spinor describes a particle with spin 1/2 and an antiparticle with spin 1/2.

To see this, the most simple and consistent approach is quantum field theory, which is simple and consistent, because it's the natural language to discuss relativistic QT, which usually deals with situations, where the particle number is not conserved, because you consider scattering processes with energy exchanges in the order of the mass of the particles and larger, and thus with some probability in the reactions particles can be destroyed and new ones can be created.

To define what describes "particles" you need to consider free, i.e., non-interacting relativistic fields first. Poincare invariance then dictates how possible wave equations look like. Further the Poincare group tells you by analyzing its irreducible unitary representations that free fields obey the Klein-Gordon equation, which gives the "dispersion relation" between frequency and wave number, $\omega=\sqrt{m^2+\vec{k}^2}$ (in units, where $c=\hbar=1$). Then also you have always also solutions with $\omega=-\sqrt{m^2+\vec{k}^2}$. This defines the "modes" of the solution, i.e., plane-wave solutions. In the 1st-quantization formalism this are the momentum eigen states, and for free particles this implies the relation between energy and momentum, but this interpretation fails here, because of the negative-frequency modes.

Now help comes from field quantization (2nd-quantization formalism). All you have to do is to make the fields operator valued and decompose the fields in plane-wave solutions. To get only states of positive energy (and the energy spectrum should be bounded from below, because there should be a state of minimum energy, such that we have stable particles as observed in Nature), you have to simply write an annihilation operator in front of the positive-frequency modes and an creation operator in front of the negative-frequency modes. That's the Feynman-Stückelberg trick without the esoteric handwaving about "particles with negative energy moving backwards in time".

To the contrary this simple mathematical trick makes the theory not only having a stable ground state but also local, i.e., it let's you fulfill the microcausality condition, according to which local observables (energy denisty, momentum density, angular-momentum density) commute if their arguments are space-like separated, which guarantees the Poincare invariance of the physical quantities you can calculate using the theory (most importantly in HEP physics are the S-matrix elements which let you predict cross sections to be measured in the scattering experients at the large accelerator facilities like CERN).

Of course, in addition to momentum and energy there's also angular momentum, and as in non-relativistic QT you can have half-integer and integer angular-momentum representations. Then one of the fascinating miracles occurs: It turns out that in order to fulfill the microcausality condition and the existence of a stable ground state you must quantize half-integer-spin fields as fermionic and the integer-spin fields as bosonic quantum fields, and this is always confirmed by observation.

Now we have a clear particle interpretation for quantized free fields with the remarkable property that, just assuming microcausality (i.e., the consistence of the theory with the causality structure needed to make sense of any relativistic theory) and the existence of a ground state (stability of matter), you conclude that for every particle there should be also an anti-particle, which has the same mass as the particle and opposite charge. As a special case any kind of field also allows for "strictly neutral" particles, i.e., such particles with all kinds of charges being 0. Then the particles and antiparticles are identical (e.g., the photon as the quantum of the electromagnetic field, although there one needs to be a bit more careful, because it's a field with 0 invariant mass, where you have to analyze everything a bit different than with massive fields; e.g., you have a field of spin 1 but only 2 and not 3 polarization degrees of freedom; all this also follows from the analysis of the proper orthochronous Poincare group), and you know that necessarily half-integer-spin fields lead to fermionic and integer-spin fields to bosonic particles.

A Dirac field is not the case of a field with spin 1/2, which describes a particle (e.g., the electron) and its antiparticle (the positron), which are fermions. It has four components, because for a particle of spin 1/2 the spin component in $z$-direction (or any other arbitrary direction, but the $z$-direction is the standard choice in QT) takes two possible eigenvalues, $\sigma_z=\pm 1/2$, and there's also the anti particle which has also spin 1/2 with also two possible $\sigma_z$-eigenvalues. So you have 4 components.

 

[snoopies622]

Thanks all, I appreciate the feedback! I'm struck by how deep this subject is, especially compared to the Schrodinger equation. Unfortunately, the more I try to understand it the more confused I become. For now the only things about the Dirac equation I'm certain of are the premises from which it was derived and the fact that the wave function has four complex numbers instead of one.

I'm wondering if someone could respond to a specific question: Suppose I observe an electron at rest and find that it is in the spin up state. If I understand this much correctly, it's now in a spin eigenstate and if I measure the spin (along the same axis) again I'll get the same result. On the other hand, if someone moving past me at a constant speed observes the same electron, what will he see?

 

[PeroK]

I'm wondering if someone could respond to a specific question: Suppose I observe an electron at rest and find that it is in the spin up state. If I understand this much correctly, it's now in a spin eigenstate and if I measure the spin (along the same axis) again I'll get the same result. On the other hand, if someone moving past me at a constant speed observes the same electron, what will he see?

The separation of spin and momentum is not so clear cut in the relativistic case. For example, the plane wave solution to the Dirac equation is only an eigenstate of $S_z$ if the momentum is in the z-direction.

In general, you must look at the total angular momentum of the electron and not the spin and orbital angular momentum separately.

 

[WernerQH]

Suppose I observe an electron at rest and find that it is in the spin up state. If I understand this much correctly, it's now in a spin eigenstate and if I measure the spin (along the same axis) again I'll get the same result. On the other hand, if someone moving past me at a constant speed observes the same electron, what will he see?

There is no such thing as an electron "at rest". The eigenvalues of the velocity operator are $\pm c$, and the electron always moves at the speed of light (in any direction). It can be "at rest" only on average. (This is called "Zitterbewegung").

In his book "The Road to Reality" Roger Penrose advocates the chiral representation, as opposed to the standard representation that you have encountered. Since angular momentum is conserved, an electron "at rest" constantly has to flip between a right-handed and a left-handed state as it is moving back and forth. An observer in a different reference frame will see a different mixture of these states. What the Dirac equation is telling us is that you cannot really separate the positive and negative energy states. Although some people frown upon the imagery, it can even move backwards in time (being called a positron then) and surround itself with virtual pairs ("vacuum polarization").

 

[vanhees71]

The standard formulation of modern relativistic local QFT avoids all this weirdness with particles moving backward in time, "zitterbewegung", etc. by just using 2nd quantization and assume microcausality as well as the boundedness of the Hamiltonian from below (stable ground state). For free particles this leads to a decomposition of the quantum fields in positive and negative frequency modes. To have a positive semidefinite Hamiltonian (you can always subtract constant and divergent "vacuum energy" contributions such that the energy of the ground state is 0) you write an annihilation operator in front of the positive-frequency modes and an annihilation operator in front of the negative-frequency modes.

Then you assume equal-time commutation (bosons) or anticommutation relations (fermions) for the field operators in such a way that the local observables (densities of energy/Hamiltonian, momentum, and total angular momentum) obey the microcausility principle, leading to the spin-statistics theorem, i.e., half-integer-spin fields have to be quantized as fermions, integer-spin fields as bosons.

In addition it turns out that the proper orthochronous Poincare group is realized by the corresponding operators (total energy and momentum, angular momentum and boost generators) derivable from the corresponding Noether charges of this symmetry group. Operating with these symmetry-transformation operators leads to a transformation behavior which looks as the linear local realization on the corresponding classical relativistic fields.

Another feature is the CPT theorem, i.e., any realization of the proper orthochronous Poincare group also admits a realization of the charge-conjugation transformation, spatial reflections, and time reversal such that applying all three of them, "CPT", is necessarily a (discrete) symmetry of the QFT. Any of the transformations P, T, C,CP, CT, PT is empirically proven to be violated by the weak interactions, but all CPT tests confirm this symmetry.

This very general properties together with the success of the Standard Model is a strong indication for the concept of local quantum field theories, and none of the old apparent paradoxes are needed to describe the behavior of the elementary particles and the matter formed by them.