Raymond Frederick Streater
Curator and author
Articles sponsored or reviewed
http://www.mth.kcl.ac.uk/~streater/ The Theorem on Spin and Statistics.
I. Introduction
Einstein's theory (1) of the photo-electric effect requires that a particle of light, later called a photon, can have any momentum, and must be in one of two polarization-states, later identified with its spin. It became clear from Bose's paper on Einstein's work (2) that the statistics of photons obey a new law; not only are two photons of the same spin and momentum indistinguishable from each other, but they are the same state. Einstein (3) expressed this by requiring that the state on n photons should be symmetric under the permutation group \(S_n\). To get the (nearly) correct energy levels for the hydrogen atom from Heisenberg's new quantum mechanics, Pauli (4) imposed the exclusion principle on electrons: each energy-level is either empty, or contains one electron: no state can contain more than one electron of a given momentum and spin. Theories of many photons, led workers to the introduction of creation operators \(a_j^*\) for a photon in the normalised state \(j\) and its hermitian conjugate, the annihilation operators \(a_j\). The physicists postulated commutation relation, \(a_ja_k^*-a_k^*a_j=\hbar i\delta_{jk}\), with all \(a_j\), (and hence all \(a_j^*\)) commuting among themselves. Then they combined this with the evident existence of a no-particle state, the vacuum, \(\Psi_0\) with the property \(a_j\Psi_0=0\), to get the correct Planck law for the distribution of the numbers of photons of various frequencies in a hot body. For electrons, the above theory does not work. They have spin of 1/2 in units of \(\hbar\), and the exclusion principle can be satisfied if the space of \(n\) electrons is deemed to be totally anti-symmetric under the group \(S_n\). Jordan and Wigner (5) introduced creators and annihilators for electrons by operators \(b_j^*,b_j\) obeying the anti-commutation relations \(b_j^*b_k+b_kb_j^*=\hbar \delta_{jk}\), where now the operators \(b_j\) anti-commute thus: \(b_jb_k+b_kb_j=o\). We shall use the usual notation $[A,B]$ for the commutator $AB-BA$ of the two operators $A$ and $B$, and the notation $\{A,B\}$ for their anti-commutator $AB+BA$. Then the use of anticommmutation relations for the creation and annihilation operators, acting on a vacuum state \(\Psi_0\), leads to antisymmetric wave-functions for all states of two or more particles, agreeing with the Pauli exclusion principle. There seemed to be no reason why there was this difference between particles of energy and particles of matter, if we leave out the principle of special relativity.
In 1927, Dirac (6) introduced his wave-equation for the electron; which transformed under the Lorentz transformations according to a unitary representation of the Lorentz group, and also under the translation group. This then was the equation of a relativistic particle of spin 1/2, joining the Klein-Gordon equation and the Maxwell equations for the behaviour of particles of spin 0 and spin 1 (of zero mass). Rarita and Schwinger (7) wrote down the equations for a particle of spin 3/2 in .... . In fact, the proof that all these equations belong to unitary representations of the Poincar\'{e} group (the inhomogeneous Lorentz group) was achieved by Wigner and Bargmann (8) in 1947; they also derived similar equations for particles of arbitrary spin and non-negative mass. The second quantization of these equations, for free particles, was done using commutators for particles of integer spin, and anti-commutators for particles of half-odd-integer spin. In 1939 Fierz (9), and 1940, Pauli (10), had shown that it is impossible to use anti-commutators for (free) relativistic particles of integer spin, and also impossible to use commutators for free relativistic particles of half-odd-integer spin. Thus, by 1940, Fierz and Pauli proved the spin-statistics theorem for free particles. We shall generalize these results, using the work of Burgoyne (11), Dell'Antonio (12), Luders (13) and Araki (14). The method uses the Wightman axioms, which are true for interacting as well as free fields. In Wightman theory, we cannot prove that the quantized field must obey the spin-statistics theorem; we can only rule out the wrong connection. In particular, the existence of parastatistics is not allowed, although there is no reason to exclude them (15). For different fields, it is normally assumed that different fields of integer spin commute at space-like separation, different fields both of half-odd-integer spin anti-commute at space-like separation, and a field of half-odd-integer spin commutes at space-like separation with a field of integer spin. Again, this assumption cannot be proved from the Wightman axioms, but it can be proved that if we assume ``abnormal relations, then certain expectations are zero, and fields obeying the normal relations can be obtained as a Klein transformation of the given fields.
A deeper theory of particle statistics is obtained by adopting the Haag-Kastler axioms (16) for the observable fields, assumed to satisfy the commuation property for observables separated by a space-like vector. Then parastatistics arise from looking at all representations of the observable algebra (17).
II. The Analytic Properties of Wightman Functions. We shall assume that a theory of elementary particles uses a quantized field, or a set of quantized fields, to describe the quantum operators of the theory. These are generalized functions of space and time; we adopt the spirit of Laurent Schwartz and assume that to each infinitely differentiable function $f$ of space and time, of compact support, is given an operator, denoted by $\phi(f)$, on a Hilbert space, $\mathcal{H}$. Wightman (18) required that the map $f\mapsto\phi(f)$ should be linear, so we can imagine that we can write
\begin{equation} \phi(f)=\int_{{\br R}^4}\;\phi(x)f(x)d^4x, \end{equation} for some generalized operator-valued object $\phi(x)$ of the space-time point $x=({\bf x},t)\in{\bf R}^4$. Wightman thus generalizes the idea of the Schwartz distribution to operator-valued distributions (19). To get a viable theory, Wightman assumes that $\phi(f)$ obeys a set of axioms, called the Wightman axioms (19). For example, it is postulated that there exists a unit vector $\Psi_0$ in $\mathcal{H}$, unique up to a phase, which is invariant under a unitary group $U(a,\Lambda)$ representing the Poincar\'{e} group up to a phase: \begin{equation} U(a,\Lambda)U(a',\Lambda^\prime)=\pm U(a+\Lambda a',\Lambda\Lambda^\prime), \end{equation} where $a,a'$ are space-time translations, and $\Lambda$ and $\Lambda^\prime$ are Lorentz transformations of space-time. To allow for the violation of parity, and time-reversal, we assume that these relations hold only for space and time non-reversing Lorentz transformations. Wightman assumes that the field, $\phi$, transforms according to a finite-dimensional representation $S$ of the Lorentz group: \begin{equation} U(0,\Lambda)\phi_\alpha(x)U^{-1}(0,\Lambda)=S_{\alpha,\beta}(\Lambda)\phi_\beta(\Lambda^{-1}x). \end{equation} Indeed, it can be proved that if $S$ is unitary (and therefore of infinite dimension), one can violate the usual relation between spin and statistics (20). Thus, we must admit that the proof of the spin-statistics theorem using the Wightman axioms is not complete; we do not prove that integer-spin fields must commute, and half-odd-integer fields must anti-commute, at space-like separated points. Rather, we assume the wrong statistics, and get a contradiction. In the algebraic approach, on the other hand, we assume that observable quantities commute (at space-like separated points}, and then obtain the existence of fermions, and para-statistical objects, obeying the usual symmetry prooperties (17). From now on, we shall adopt the Wightman approach to the problem.
The uniqueness of the vacuum vector up to a complex multiple implies the cluster decomposition theorem for the Wightman functions, and conversely {21). This implies the relation, along a space-like four-vector $a$, \begin{equation} \lim_{\lambda\rightarrow\infty}\langle\Psi_0,\psi(f)^*\psi(f)U(\lambda a)\varphi{g}^*\varphi(g)\Psi_0\rangle\rightarrow\langle\Psi_0, \psi(f)^*\psi(f)\Psi_0\rangle\langle\Psi_0,\varphi(g)^*\varphi(g)\Psi_0\rangle. \end{equation}
We shall need the Reeh-Schlieder theorem, which follows from the Wightman axioms, when we assume, for each pair of components of the field, rhat either $[\varphi(x),\psi(y)]$ or $\{\varphi(x),\psi(y)\}$ vanishes when $x-y$ is a space-like vector. This says that for any bounded open subset ${\cal O}$ of ${\bf R}^4$, the set of vectors of the form $\varphi(f_1) ... \psi(f_n)\Psi_0$, $n=0, 1,2, ...$, span the Hilbert space of the theory, when all fields are included, but all the functions $f_1, ... , f_n$ are zero outside ${\cal O}$: the vacuum is a cyclic vector for any local algebra. It is a consequence of this theorem that the vacuum is also separating for any local algebra: if $\psi(f)\Psi_0=0$ for some $f$ of compact support, then $\psi(f)=0$.
III. Spin and Statistics for a single spin-multiplet of fields
We first show that in a theory with a unique vacuum, a field-component $\varphi$ which commutes at space-like separation with a field $\psi$ cannot obey anti-commutativity at space-like separation with the hermitian conjugate field $\psi^*$. A similar argument shows that if $\varphi$ anti-commutes with $\psi$ at space-like separation, then it cannot obey commutativity at space-like separation with the conjugate $\psi^*$. This idea was proved in (12).
Theorem 1. In a quantum field theory with a unique vacuum, the requirement that both $[\varphi(x),\psi(y)]=0$ and $\{\varphi(x),\psi^*(y)\}=0$ for all spacelike $x-y$ implies that either $\varphi=0$ or $\psi=0$.
Proof. Let $f$ and $g$ be test-functions of compact suppport; then we have \begin{equation} \langle\Psi_0,\varphi(f)^*\psi(g)^*\psi(g)\varphi(f)\Psi_0\rangle=\|\psi(g)\varphi(f)\Psi_0\|^2\geq0. \tag{1} \end{equation} Now suppose that the supports of $f$ and $g$ are space-like separated. The assumed commutation and anticommutation relations for the fields then imply that the left-hand side of Eq.~((1)) is \begin{equation} -\langle\Psi_0,\psi(g)^*\psi(g)\varphi(f)^*\varphi(f)\Psi_0\rangle.\tag{2} \end{equation} Now for fixed $f$ translate $g$ to infinity in a space-like direction. Then by the cluster decomposition theorem, which holds because the vacuum is unique, the expression ((2)) converges to \begin{equation} -\langle\Psi_0,\psi(g)^*\psi(g)\Psi_0\rangle\,\langle\Psi_0,\varphi(f)^*\varphi(f)\Psi_0\rangle=-\|\psi(g)\Psi_0\|^2 \|\varphi(f)\Psi_0\|^2 \end{equation} and this is not positive. Comparing with Eq.~((1)) the limit must be zero: either $\psi(g)\Psi_0=0$ or $\varphi(f)\Psi_0=0$. If $\psi(g)\neq 0$, then $\psi(g)\Psi_0\neq 0$, since the vacuum is separating. Then we see that $\varphi(f)\Psi_0=0$, which implies that $\varphi(f)=0$. Thus either $\psi=0$ or $\varphi=0$. \Box\(Insert formula here\)
References. (1) (2) (3) (4) (5) (6) (7) (8) (9) M. Fierz, Uber die relativische Theorie kraftfreier Teilchen mit beliebigem Spin, Helv. Phys. Acta, 12, 3, 1939 (10) W. Pauli, On the Connection of Spin with Statistics, Phys. Rev., 58, 716, 1940 (11) N. Burgoyne, On the Connection of Spin with Statistics, Nuovo Cimento, 8, 153, 1958 (12) G. F. Dell'Antonio, On the Connection of Spin with Statistics, Annals of Physics, 16, 153, 1961 (13) G. Luders, Vertauschungsrelationen zwischen verschiedenen Feldern, Z. Naturforsch. 13a, 254, 1958 (14) H. Araki, Connection of Spin with Commutation Relations, J. Mathematical Phys, 2, 267, 1961 (15) O. W. Greenberg and A. Massiah, Are there Particles in Nature other than Bosons or Fermions?, Phys. Rev., 136, B248, 1964 (16) R. Haag and D. Kastler, An Algebraic Approach to Quantum Field Theory, J. Math. Physics, 5, 848-861, 1964 (17) S. Doplicher, R. Haag and J. Roberts, Local Observables and Particle Statistics, Commun. Math. Phys. 23, 199-230, 1971. (18) Wightman, A. S., Quantum Fields in Terms of their Vacuum Expectation Values, Phys. Rev. ,860-, 1956. (19) Streater, R. F. Wightman Quantum Field Theory, Scholarpedia. (20) Streater, R. F., Local Fields with the Wrong Relation between Spin and Statistics, Commun. Math. Phys. 5, 88-96, 1967. (21) Hepp, K., Jost, R., Ruelle, D., and Steinmann, O. Necessary Conditions on Wightman Functions, Helvitica Physica Acta 24, 542, 1961.
