Why high-spin particles are not seen yet?
(Some
derivations have been removed since it's not easy to input mathematical
equations here. To see the full text please download the PDF file
attached. Here is the LINK.)
Introduction
Historically, the concept of spin was introduced by Uhlenbeck and Goudsmit in 1925, in order to explain somewhat weird result of the well-known Stern-Gerlach experiment. They hypothesized that, every electron has an intrinsic angular momentum of \hbar/2. At that time, however, the origin of this intrinsic angular momentum was not clear. Naturally, one might identify the spin of an electron as the rotation along the axis passing through its center. But this does not work, as posed by Lorentz, who showed that the linear velocity of the “surface” of an electron will exceed the speed of light, if such a viewpoint is taken. This is evidently forbidden by the theory of relativity.
The rigorous and systematical treatment of the theory of spin was first given by Wigner, who developed his theory in the frame of the quantum mechanics. As we know, the central idea of the quantum mechanics is the quantum state and the Hilbert space. If a particle can be represented by a state in the Hilbert space, then the symmetry that governs the motion of the particle will also acts on the Hilbert space. We know that symmetry can be described mathematically by a group, thus the action of the symmetry on the Hilbert space can be accordingly described by the representation of the group.
The crucial thing here is that the representation of a symmetry group on a physical Hilbert space must be a unitary (or anti-unitary) representation. This is the famous Wigner theorem. A direct consequence of this theorem on particles, is the fact that a massive particle with spin s has 2s+1 degrees of freedom, while a massless particle always has two degrees of freedom, which has nothing to do with its spin. In principle, the spins of both massive and massless particles can take any positive integer and half-integer value, including zero.
On the other hand, in quantum field theory, particles are created by field operators, which can be classified by their transformation properties under Lorentz transformations. The different classes of fields are known as scalar, vector, or tensor, etc. Of course, they are also the representations of the symmetry group of the space-time, but these representations are quite different from ones carried by states. Since the former is finite-dimensional and non-unitary, while the latter is infinite-dimensional and unitary. This fact leads to a problematic result: the degree of freedom (DOF) of the field will in general be different from the DOF of the state (or particle) created by that field. To fully understand this problem, we will introduce two interesting theorems. They are known as “no-go” theorems which mean the statement of the theorems are negative.
Weinberg-Witten Theorem
The Weinberg-Witten theorem mainly deals with the massless particles. The formal statements of the theorem are as follows:
Theorem 1: A theory that allows the construction of a Lorentz-covariant conserved four-vector current J^\mu cannot contain massless particles of spin j>1/2 with nonvanishing values of the conserved charge \int\di^3x J^0.
Theorem 2: A theory that allows the construction of a conserved Lorentz covariant energy-momentum tensor T^{\mu\nu} cannot contain massless particles of spin j>1.
The proof of the theorem is straightforward. The strategy is to consider the S-matrix elements of the conserved current.
(To see the details of the proof, please download the PDF file attached.)
Coleman-Mandula Theorem
The Weinberg-Witten theorem excludes the presence of charged massless particles with too large spin. However it says nothing on massive particles. Now we introduce the more powerful Coleman-Mandula theorem, which is also a no-go type theorem.
Theorem
1) For any M there are only a finite number of particle types with mass less than M.
2) Any two-particle state undergoes some reaction at almost all energies.
3) The amplitude for elastic two-body scattering are analytic functions of the scattering angle at almost all energies and angles.
With these assumptions, the theorem claims that the only possible Lie algebra of symmetry generator consists of the generators of the Poincaré group, together with possible internal symmetry generators, which commute with the Poincaré generators.
A possible explanation of the absence of high-spin particles
With the Coleman-Mandula theorem in hand, let us go back to the problem of the spin. As has mentioned in Section 1, the degrees of freedom between the field and the corresponding state have a nontrivial mismatch when the state has the spin s≤1. For instance, A vector field, which has 4 DOFs, can create a state with spin 1, which has only 3 (or 2 in massless case) DOFs. Another example is the gravity: A metric field has 10 DOFs, while a graviton, as a massless particle, has only 2 polarizations.
We see that as the spin goes higher, the mismatch between fields and states becomes more serious. This result suggests that there exist redundant and unphysical DOFs in fields. To exclude these redundant DOFs, we should impose the gauge symmetry on the fields. Conventionally, these kinds of fields are called gauge fields. It explains why gauge symmetry is necessary.
As we have learned in classical electrodynamics, in a physical theory with gauge symmetry, the gauge field must couple to a conserved current to maintain the gauge invariance. Generally, we can write this coupling term in the Lagrangian as:
(Omitted derivations)
Now the Coleman-Mandula theorem works: The theorem claims that all the conserved charges, or generators of inner symmetries commute with Lorentz generators, hence these charges Q must be scalars and carry no Lorentz indices. Then the current corresponding to such a generator must be a vector J^\mu. so as the field coupled to the current. We conclude that inner symmetries can only offer couplings to a vector fields, which corresponds the spin 1 particle.
The remaining choice of the generators are Lorentz generators. For example, the momentum generator P^\mu, as a vector, produces a conserved current of rank-2 tensor T^{\mu\nu}, which is just the well-known energy-momentum tensor. This tensor couples to gravity, thus make an opportunity for us to detect the spin-2 gravitons. The last choice is angular momentum generator J^{\mu\nu}, which permits a coupling to a rank-3 tensor field, which I haven't heard about yet.
Now the list of symmetry generators is exhausted. We see that no elementary particles with spin higher than 3 can be detected, due to the lack of proper type of interactions.
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它来自Aimard在Teldec的唱片：
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