Particle Lagrangians

Question

As I learned in my string theory course, you can describe the quantum spin-0 particle by quantizing the arclength Lagrangian of a relativistic classical particle.

My question is whether you can get a photon, or arbitrary particle with spin, by quantizing a particle Lagrangian. Since we usually call our quanta in quantum field theory "particles," it seems we should have a Lagrangian describing said particles.

I have heard of some constructions involving worldline supersymmetry. What is the relation between worldline and spacetime supersymmetry in such Lagrangians? Is it possible to have a Lagrangian without worldline or spacetime supersymmetry?

If anyone knows a good reference or review article I would be most happy!

Answer 1

Yes, you may obtain spin from first-quantized Lagrangians as well. However, such a particle with spin must have some extra degrees of freedom - which don't need to be field-theory degrees of freedom.

In supersymmetric theories, one may consider the propagation of a point-like particle in a superspace instead of a regular bosonic space. The world line of such a "superparticle" is then given by functions $$X^\mu (\tau), \theta^a (\tau)$$ where $\theta$ are additional Grassmannian (fermionic) functions of the parameter $\tau$ along the world line. Once you consider the quantum version of this theory, you will find out that the wave function of the particle depends not only on $X$ but also the fermionic directions $\theta^a$. The wave function may be Taylor-expanded in these $\theta$ variables - one only gets $2^N$ terms where $N$ is the number of these $\theta$ variables. And the individual coefficients in front of products of $\theta$ describe the wave function for the particle's being in a particular state of spin, too.

For example, you may consider a superparticle in 11 dimensions of M-theory. Then you effectively have 8 $\theta$ variables that matter - after you eliminate most of the covariant 32 components of the corresponding spinor. The Taylor-expanded function of the 8 physical $\theta$ variables has 256 components and they describe various components of the graviton field, gravitino, and the $3$-form generalized gauge potential.

All of string theory is another major example of what you're asking. It's because perturbative string theory is usually studied in the first-quantized form. Aside from the "center of mass" degrees of freedom $X^\mu$ and their possible fermionic (supersymmetric) counterparts $\theta^a$ I discussed above, a string is a particle that also has the other Fourier modes $X^\mu_n$ and $\theta^a_n$ where $n$ is an integer - they can be obtained by Fourier-transforming $X^\mu(\sigma)$ and $\theta^a(\sigma)$.

The $n\neq 0$ modes of $X$ and $\theta$ are internal degrees of freedom of the particle and by exciting them, in the same way as we do for the harmonic oscillator etc., we may change the spin of the particle as well. Some fermionic degrees of freedom are always necessary for the spin to be allowed to take half-integral values.

Alternatively, string theory may be formulated as a generalized field theory, the so-called "string field theory". One may only calculate physically equivalent answers from this "second-quantized approach" to string theory and because it doesn't bring us too many new things, it's only studied by a rather limited group of string theorists who are fascinated by the formalism of string field theory itself.

Answer 2

I want to add to Lubos answer and give some old and new references on the classical representation of spinning particles. In addition, I'll try to explain, the main idea of the construction.

The classical description of spin by means of Grassmann variables was introduced by F.A. Berezin and M.S. Marinov (for the nonrelativistic case) in: F. A. Berezin and M. S. Marinov, Classical spin and Grassmann algebra, Sov. Phys. JETP Lett. 21 (1975) 320-321. (This article can be found in Prof. Marinov memorial page).

Later, this work was extended to the relativistic case in: F.A.Berezin and M. S. Marinov, Particle spin dynamics as the Grassmann variant of classical mechanics, Ann. Phys. (NY) 104 (1977) 336-362. (Here is the version of this article).

The relativistic Lagrangian is given in this paper is in the arc length (square root) form.

I'll try to briefly explain the logic behind their construction:

The translational phase space is parameterized by the position $q_i$ and momentum $p_j$ coordinates which are:

1. Commuting as functions on the phase space $q_jq_i=q_iq_j, p_jq_i=q_ip_j, p_jp_i=p_ip_j$
2. Satisfying the canonical commutation relation: $ \{q_i,q_j\} =0, \{p_i,p_j\} =0, \{q_i,p_j\} = \delta_{ij}$
3. Their quantization operators satisfy, (the Weyl algebra) commutaion relation $[\hat{q}_i,\hat{p}_j] = i \hbar \hat{1} \delta_{ij}$.

In analogy, the spin phase space is parameterized by the Grassmann variables $\xi_i$, which are:

1. Anti-commuting as functions on the phase space $\xi_j\xi_i=-\xi_i\xi_j$
2. Satisfying the commutation relation: $ \{\xi_i,\xi_j\} = \frac{1}{2i}\delta_{ij}$
3. Their quantization operators satisfy, (the clifford algebra) commutaion relation $[\hat{\xi}_i,\hat{\xi}_j] = \frac{\hbar}{2} \delta_{ij}$.

Thus, the main idea is that as the Weyl algebra is the quantization of the classical translational phase space, the Clifford algebra is the quantization of the classical spin phase space.

As known from the theory of Clifford algebras, the spin generators are the bivectors in the basic Clifford generators, for example in the three dimensional case:

$\hat{S}_i = i \epsilon_{ijk} \hat{\xi}_j\hat{\xi}_k$.

In the infinite dimensional case, one obtains, the usual the canonical anti-commutation algebra (CAR).

A review article by Andrzej Frydryszak of the Berezin-marinov model can be found in: LAGRANGIAN MODELS OF PARTICLES WITH SPIN: THE FIRST SEVENTY YEARS

It should be mentioned that the Berezin-Marinov model describes only spin $\frac{1}{2}$ particles. (This property is desirable from a quantization theory, because it results an irreducible representation of the operator algebra). In order to describe photons, one should add additional noncommuting coordinates to the phase space, See for example the following solution by : Gitman and Goncalves

Finally, the Berezin-Marinov construction is a prototype geometric quantization model of a phase space which is symplectic superspace, namely the supercotangent bundle of the bosonic phase space. The following article: Conformal geometry of the supercotangent and spinor bundles: by J.P. Michel describes the model from this point of view and its generalization to the non-flat case, where the configuration space is a Riemannian manifold. Here, the quantization of the supercotangent bundle of a spin manifold is its spinor bundle.

Answer 3

As spin has no classical analog, you either need to start with a classical field theory, or a quantum wave function.