# Spinors in QFT and QM

1. Dec 3, 2014

### HomogenousCow

Often I see QFT texts introduce dirac spinors by comparing them to the two component spin states (which I have come to accept are also spinors) in NRQM. And arguing that since the NRQM spinors transform via SU(2), our desired quantum fields for spin 1/2 particles should be some higher dimensional version of this.

I don't quite see the logic here. The NRQM spinors are states while the dirac field is basically an operator (or a bunch of them).
What am I missing? There seems to truly be some connection yet I cannot wrap my mind around it.
Spin comes out of the dirac spinor field as a conserved current due to the field itself transforming under lorentz transformations.
Spin in NRQM is kind of just ad-hoc put in by hand.

2. Dec 3, 2014

### Demystifier

You are missing the difference between classical fields and quantum fields. Quantum fields are operators. Classical fields are not operators, so can be thought of as wave functions (i.e. "states"). Both classical and quantum fields may have either 2 components (e.g. in non-relativistic limit) or 4 components. The presence of 2 components usually indicates that you describe only the electron degrees of freedom, while the presence of 4 components usually indicates that you describe both electron and positron degrees of freedom.

3. Dec 3, 2014

### WannabeNewton

No that's not true at all. In both QM and QFT spin comes from representations of rotation and spinors are, in the usual context, objects that transform under the $j = 1/2$ representation. The main differences in QFT are: the representations are field representations; one can have both kinds of representations of rotation or other Lie groups; we want Lorentz invariance so we use the double-covering which leads to a Lie algebra that breaks up into those of two copies of $SU(2)$ and this leads to Dirac spinors as opposed to just Weyl spinors.

Last edited: Dec 3, 2014
4. Dec 3, 2014

### HomogenousCow

That makes sense to me.
However the process of writing down a classical lagrangian first seems so different from the whole thing with spin in NRQM, I guess the formalism is making me confused.
However, wouldn't this line of thinking be more direct?
Let's say we already know about scalar fields and charged scalar fields, but they have no spin so we want to find charged spin fields.
We know that real representations of the lorentz group don't have charge because of no U(1) symmetry.
So we look at more complicated complex representations.

5. Dec 3, 2014

### Demystifier

Good!

Yes, that's a little confusing. To see more confusions of this sort in quantum theory see

Yes, it would make sense.

6. Dec 3, 2014

### HomogenousCow

I have an unrelated question that I would like to ask.
What do people mean exactly when they say that nucleons are made of quarks?
I'm made to understand that quarks are dirac spinors, so what does it mean to bind three (or two) of these guys together?
Is this done directly in the Lagrangian or is it some emergent behavior whereby the gluon field binds triplets together.

7. Dec 3, 2014

### The_Duck

Something similar to what they mean when they say "a hydrogen atom is made of a proton and an electron."

Same as atoms with electromagnetism. Atoms are bound states of nuclei and electrons held together by the electromagnetic field. Nucleons are bound states of quarks held together by the color field.

8. Dec 3, 2014

### dextercioby

The concept of spin and the concept of spinor (<spinor vector/tensor, spinorial field) are related, but rather different. Spin is an observable (<spin angular momentum), a set of operators in quantum mechanics. Spinors are correctly quantum fields (in the Wightman sense), or (incorrectly or better say, totally invented (see post#9 below) classical fields (in the sense of GR, for example, but with SU(2) instead of SO(3) and SL(2,C) instead of SO$_{\uparrow}$(1,3), whose use really makes no sense outside quantum mechanics for the known reasons).

Digression (hope it makes some sense to some people-it ends before the quote below :))
<We cannot picture spin in the absence of spacetime, be it Galilean, Minkowskian or even curved. Spin is a consequence of symmetry (more precisely the symmetry group of dynamics of the underlying spacetime) as pointed out (to my knowledge) by a lot of people, among the last of them being T.F. Jordan towards the end of the 1960s (the work of T.F. Jordan inspired L. Ballentine when he first wrote his QM textbook in 1989). As a consequence (of the explainable concept) of spin , we have spinors (spinorial fields). So what's a spinor (field)? (to my shame I really have zip knowledge of the diff.-geom. intricacies involved). Geometers tell us: A section of a spinor bundle over space-time, as R. Wald teaches us in his chapter 13 of his (marvelous) 1984 GR book (pages 365-366). So just as there's no spin without symmetries of spacetime, there are no spinors without spacetime, and definitely no spin, nor spinors without some hardcore representation theory of Lie groups (in a very interesting conjunction with functional analysis and differential geometry). But in a very symplistic manner (i.e. for example leaving the theory of fiber bundles aside), we can define spinors as (classical/quantum) objects (set of mathematical functions) covariant with respect to the (restricted) Galilei/Minkowski group. Examples: A scalar field (such as the temperature field in a Galilean space-time) is a trivial (rank-0) Galilean spinor field. A vector field (such as the velocity field in an ideal (Euler) fluid) is a rank-1 Galilean spinor field. Any-rank tensor field is actually a same-rank spinor field and that's true for Minkowski/Lorentz spinor fields as well (think of the e-m Faraday tensor field).

Q: What is then a Dirac spinor?
A: A spinorial quantum field (as opposed to state*) in the the sense of Wightman which describes SPIN 1/2, massive, electrically charged elementary leptons + quarks.

Q: So if a Dirac spinor is a quantum field, then what's the set of 4 functions discovered by Dirac in 1928?
A: A historical accident actually, which came as the product of the numerous attempts (stretched along 2 years) to building a Schrödinger wave-mechanics in agreement with special relativity (The 1926 Schrödinger's wave-mechanics's agreement with Galilean relativity had not been discovered, much less exploited), Dirac merely had the brilliancy to linearize the KG equation. The 4 mathematical functions that Dirac found are coordinate-representation (in the sense of wave-mechanics) quantum states (kets = vectors in the sense of Hilbert spaces, or rather Rigged Hilbert Spaces).

Q: So the Dirac spinor can be considered as an archetypal example of the known dychotomy in quantum theories: states vs. observables (measurable quantities)?
A: Yes, but with a caveat: In specially-relativistic QFT the concept of observables defined in NRQM is replaced by the overwhelming concept of (quantum) fields. Empirically the Dirac spinorial field is not measurable, but neither were the 4 wavefunctions of 1928. >

That's plain wrong in so many ways.

Last edited: Dec 3, 2014
9. Dec 3, 2014

### dextercioby

Classical fields can be seen as the non-quantized verson of (Wightman) quantum fields of integer spin (that's really tautological, I know) and belong to the (from my perspective little textbook-aknowledged) discipline of classical field theory (sorry, again it's hard to avoid the tautology) which exists very well completely outside of quantum mechanics and its wave functions. The <classical fields [...] can be thought of as wave functions (i.e. "states")> is a dangerous and misleading analogy, for classical fields are actually (Lagrangian or Hamiltonian) observables, not "states" (the concept of state is actually pertaining to QM and QFT).

There are no true classical fields of 2 components, if one is considering spacetime made up of 3 spatial coordinates and 1 time coordinate. You're probably thinking of the <dequantized> Weyl fields, but they are not classical, as classical field theory is done with finite dimensional representations of SO(3) Galilei and SO(1,3)-Lorentz only. They are (as I said above) 'invented'.

Last edited: Dec 3, 2014
10. Dec 4, 2014

### Demystifier

Yes, I have already noted something in that spirit in my "confusion thread" linked above in post #5.

11. Dec 4, 2014

### vanhees71

I think, here is some confusion about "spin" in this discussion. First of all the spin of a particle, $s \in \{0,1/2,1,3/2,\ldots \}$ defines the representation of the rotation group, which is a subgroup of both the Galilei symmetry of Newtonian space-time and Minkowski symmetry of special relativistic (Einsteinian) space-time.

Let's discuss the relativistic case right away since this was asked in the OP and let's concentrate on the massive case.

More specifically for massive particles, the spin is defined as the representation of the rotation group for a particle at rest in a given inertial frame of reference. This is the socalled "little group" of the massive representations of the unitary representations of the covering group of the special orthochronous Poincare group. Covering group means that in the semidirect product of translations and $\mathrm{SO}(1,3)^{\uparrow}$ the special orthochronous Lorentz group is substituted by its universal covering group, which is $\mathrm{SL}(2,\mathbb{C})$. The representation of the little group induces the representation of the full group via the Frobenius construction (see Wigner 1939).

Now there are a special type of representations of the Poincare group in quantum field theory, namely those with local field operators, i.e., where you have field operators that transform unitarily under the Poincare group such as the analogous classical fields. This concept of locality is very important, because it's sufficient to construct microcausal theories, which lead to a unitary Poincare covariant S matrix, which defines the observables of quantum field theory like decay rates and cross sections (I omit the idea of observables within relativistic many-body theory, which is a quite straight-forward extension of the observables defined via the S matrix in vacuum QFT). Together with the boundedness of the Hamiltonian these concepts lead to the very successful QFTs describing the Standard Model of Elementary Particles and effective relativistic QFTs (e.g., to describe low-energy QCD in terms of (resummed) chiral perturbation theory for hadrons).

This very abstract mathematical considerations justify a shortcut, which is very valuable to introduce relativistic QFT to beginners in the field and coming quickly to the physical applications: canonical field quantization. This is a heuristical method to get to the QFTs underlying the standard model more quickly. The idea is to look for local Poincare invariant action functionals. Local means the action functional is derived from a Lagrangian that is a polynomial of fields and its first space-time derivatives such that the action functional is Poincare invariant. The classical field equations are then Poincare co-variant, where the Poincare symmetry is realized as (not necessarily in any sense unitary) local representations. E.g., for a vector field, the transformation under Boosts and rotations is given by
$$A'{}^{\mu}(x')={\Lambda^{\mu}}_{\nu} A^{\nu}(\Lambda^{-1} x'), \quad \Lambda \in \mathrm{SO}(1,3)^{\uparrow}.$$
Unter translations the fields transform as scalars.

To find all possible classical fields, including those where the $\mathrm{SO}(1,3)^{\uparrow}$ is substituted by its covering group $\mathrm{SL}(2, \mathbb{C})$, it is sufficient to investigate the finite-dimensional linear transformations of this group (there are no unitary ones except the trivial one since $\mathrm{SL}(2,\mathbb{C})$ is not compact) or its Lie algebra. This Lie algebra is isomorphic to a direct sum $\mathrm{su}(2) \oplus \mathrm{su}(2)$, but it's complexified. The rotations, as a subgroup of the Lorentz group can be represented unitarily with finite-dimensional representations, because it's a compact Lie algebra (all irreducible representations (irreps) are isomorphic to a unitary one). Due to the structure as a direct sum of two su(2) algebras, each irrep is characterized by two integer or half-integer numbers $(s_1,s_2)$, each giving the representation for the two Lie algebras. For the rotation group as a subgroup, the representations are in general not irreducible, and to define elementary particles of definite spin, one has to project out all irreducible parts except the one you want to use to describe an elementary particle. The irreps. of the SU(2) as a subgroup are given by the rules of angular-momentum addition ("Clebsch-Gordan gymnastics"), i.e., it contains spin representations for $s \in \{|s_1-s_2|,|s_1-s_2|+1,\ldots,s_1+s_2 \}$. Projecting out the unwanted spin states for an elementary particle of definite spine is usually done with the field equations and appropriate constraints.

E.g., for a vector field you have four components $A^{\mu}$. A possible relativistically covariant equation is the Proca equation,
$$\partial_{\mu} F^{\mu \nu}=-m^2 A^{\nu}, \quad F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$$
The vector field belongs to the (1/2,1/2) representation of the Lorentzgroup, i.e., it contains a s=0 (scalar) piece and an s=1 (vector) piece for the representation of the rotation group as a subgroup. To project out the unwanted scalar piece, the field equation is already sufficient (for the massive case only!), because the Proca equation implies the constraint $\partial_{\mu} A^{\mu}=0$. You are left with three field-degrees of freedom which represent the rotations in the s=1 representation 2s+1=3 is the dimension of this representation).

Now let's come to fields representing (in the quantized version) spin 1/2. The two most simple possibilities are the representations (1/2,0) and (0,1/2), which are both 2-dimensional and only contain the s=1/2 representation for the rotation subgroup. The associated $\mathbb{C}^2$ valued fields are called Weyl-spinor fields. In the standard model of (massless Dirac) neutrinos the neutrinos are represented by such Weyl fields.

Now to the Dirac fields: They are introduced to admit an extension of the representation of the proper orthochronous Lorentz group with spatial reflections (parity). One can prove that this can be achieved (uniquely) by the direct sum $(1/2,0) \oplus (0,1/2)$. This makes a representation with two Weyl spinors, where the space-reflection trafo involves and interchange of these two Weyl spinors (the direct realization of this theorem is known as the Weyl or chiral representation of the Dirac algebra). That's why quarks and the massive leptons are represented by Dirac fields, because it admits parity conservation (space-reflection symmetry) for the strong and the electromagnetic part of the Standard Model. Only the weak interaction with its "vector-minus-axial vector" structure violates parity and thus is realized as a chiral gauge theory. Of course this simple picture is a bit complicated by the fact that in the standard model we need a realization as a Higgsed gauge symmetry leading to quantum flavor dynamics (Glashow-Salam-Weinberg model), which "mixes" (or in a certain weak sense unifies) the electromagnetic and the weak interactions.

For more on spinors and spinor fields, see my QFT manuscript

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

or Weinberg, Quantum Theory of Fields I;
Sexl, Urbandtke, Relativity, Groups, Particles.

12. Dec 4, 2014

### haushofer

Check out the 60's papers of Levy-Leblond on spin in NR-QM :)

13. Dec 4, 2014

### Demystifier

He derived Pauli equation from linearization of the Schrodinger equation. But unlike Dirac for the Dirac equation, he did not explain why one should linearize the Schrodinger equation.

14. Dec 4, 2014

### dextercioby

Where's the problem in that? That's what research is all about: creating new physics and that's what Levy-Leblond did.

15. Dec 5, 2014

### Demystifier

First, I would say he did not really created new physics, but a new derivation of old physics. That's fine, but if a new derivation uses a new assumption, then it is desirable that the new assumption looks plausible. What I am a bit skeptical about is plausibility of his assumption that the Schrodinger equation should be linearized. I am not saying that his derivation has no value; it certainly has. I am merely suggesting that the value would be even greater if one was able to give an independent justification for doing linearization.

16. Dec 5, 2014

### vanhees71

I don't see why you have reason to feel uneasy with the derivation of the Pauli equation from symmetry considerations, because precisely the same assumptions go into it when deriving the Dirac equation. The only difference is that for the Pauli equation you use the Galilei invariance of Newtonian space-time and for the Dirac equation you use Poincare invariance of Minkowski space-time.

There's one additional assumption also used in both the relativistic and the non-relativistic case, namely that you get to the correct description of the interaction of particles with an external electromagnetic field by a heuristic principle called minimal substitution, which is based on gauge invariance of electromagnetics. This is ad hoc in both derivations and only justified through the tremendous success of this principle in describing electromagnetic phenomena, including the spectral lines of atoms. Of course, the relativistic derivation feels more consistent, because electromagnetism is by its very nature a relativistic theory, but it is also true that the Pauli equation can be systematically derived from the Dirac equation in a systematic way as an expansion of the Hamiltonian in powers of $1/c$, where $c$ is the speed of light in vacuo.

The far more important point is that the Dirac equation, which is relativistic, does not admit a single-particle wave-function interpretation without getting a conflict with microcausality and thus is necessarily a many-body framework as is any relativistic quantum theory, while there's no such problem with the Pauli equation in the realm of non-relativistic QT.

17. Dec 5, 2014

### Demystifier

You missed my point entirely, but that's probably because I have not explained it explicitly. So let me be explicit.

Dirac's reasoning: Klein-Gordon equation contains a second time derivative, so the time-component of the conserved current is not positive, so it cannot be the probability. To cure this, let me replace the Klein-Gordon equation with an equation with a first time derivative.

Levy-Leblond reasoning: Schrodinger equation is already a first time-derivative equation, so it does not have a problem with probability. Nevertheless, let me replace the second space-derivatives with first space-derivatives. But why?

18. Dec 5, 2014

### haushofer

I see your point now also. I have to think about that :)

19. Dec 5, 2014

### vanhees71

Ironically this first heuristic motivation by Dirac to establish a first-order-in-time relativistic wave equation is also flawed, and we know very well nowadays way, and that's the true merit of Dirac's work on his equation: For interacting particles (and only those are observable and thus make sense physics wise), there is no sensible interpretation of any formalism of relativistic QT in terms of single-particle wave functions which works so beautifully for the position representation of non-relativistic quantum theory. The reason is that there is no known interaction which keeps particle number conserved in relativistic theories. The conserved intrinsic quantities are charges.

Originally Dirac wanted to describe electrons, interacting via the electromagnetic interaction, in a way as this was done by Schrödinger with his single-particle wave mechanics. Thus he thought, in order to get a conserved positive definite "Noether charge", he could succeed using a first-order wave equation as in the Schrödinger equation. The analysis, how to set up such an equation led him via the now familiar formalism of using the Clifford algebra of the Minkowski pseudometric to his Dirac-spinor formalism. First of all he recognized that now he describes spin-1/2 particles, and indeed he found that the Noether current from the global phase invariance is positive definite. This looked promising.

However, his Hamiltonian was not bounded from below and thus the system would have no stable ground state. Now Dirac's really intuitive insight came: He considered the spin-1/2 particles to be fermions (sure, electrons are indeed fermions after all) and he thought that the ground state is given by the state, where all "negative-energy states" are filled with electrons. Then a single electron is described as an excited one above this "Dirac sea", and then it might happen that at very large collision momenta one might kick out an electron from the sea leaving a hole. This means "lack of negative energy" and "lack of one electron charge". In conclusion since the filled sea represents the vacuum, what's described with this idea then is that the hole appears as a positively charged particle with positive energy. In this way his model made sense at the prize predicting a new particle, which of course was found later and is known as the positron (antielectron).

Now, this is a contradictio in adjecto: Starting with the aim to create a single-particle theory consistent with relativistic physics he was first to introduce a very artificial many-body system, with the vacuum being filled with an infinite sea of electrons. It turned out that one, however, can work out QED in this "hole-theoretical formulation", but it's very cumbersome and not so convincing.

Nowadays we formulate relativistic QT as local microcausal quantum field theories, and the Dirac field becomes quantized as a fermion (quantizing it as bosons would again lead to a theory without stable ground state; which is a special case of the spin-statistics theorem, according to which half-integer-spin fields have to be quantized as fermions and integer-spin field as bosons). Then, after coupling photons to the theory, one recognizes that the conserved four-current from the phase symmetry does not describe particle density but electric-charge density with the electrons carrying 1 negative and the positron carrying 1 positive elementary charge. The current operator has to be defined via the Stückelberg formalism and normal ordering has to be introduced to make sense of the mathematically ill-defined field-operator products at the same space-time point. This gives a consistent picture of QED in a relativistic QT framework, which always is a many-body system to begin with, and indeed nowadays it's everyday business to produce and destroy particles in high-energy experiments like the LHC :-).

20. Dec 5, 2014

### Demystifier

I absolutely agree. Today we can use the theory of representations of the symmetry group to derive the correct wave equation with a spin, either relativistic or non-relativistic. But that's not how Dirac derived the Dirac equation, and not how Levy-Leblond derived the Pauli equation.

In addition, let me note that Levy-Leblond has not mentioned his derivation of Pauli equation in his book "Quantics" in 1990. I guess he realized himself that his old derivation of Pauli equation does not look very convincing from a modern point of view.