Mathematical Physics and Quantum Field Theory, Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 207–213 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

The spin-statistics connection: Some pedagogical remarks in response to Neuenschwander’s question ^{∗}

A. S. Wightman

In 1994, D.E. Neuenschwander posed a question (American Journal of Physics 62 (1994) 972): “Has anyone made any progress toward an ‘elementary’ argument for the spin-statistics theorem?” The theorem in question was proved in 1940 by W. Pauli and says that particles of integer spin cannot be described by fields that anti-commute at space-like separated points, i.e., cannot be fermions, while half-odd-integer spin particles cannot be described by fields that commute at space-like separated points, i.e., cannot be bosons.

It seems to be characteristic of many of the attempts at an elementary proof that the authors want to use only quantum mechanics and the transformation law of the wave function under the Euclidean group or, equivalently, under translations and rotations of space. My first point is that under these hypotheses there is no Spin-Statistics Connection. To prove this statement, pick a positive real number, s, that is either an integer or half an odd integer and construct operators Ψ(x⃗, M), M = −s, −s + 1, … (s − 1), s, satisfying

V (a⃗, A)Ψ(x, M)V (a⃗, A)^{−1} = ∑_{L} D^{(s)}(A^{−1})_{ML}Ψ(R(A)x⃗ + a⃗, L)  (1)

where {a⃗, A} → V (a⃗, A) is the unitary representation of the Euclidean group giving the transformation law of the wave function. R(A) is the rotation determined by A and A → D^{(s)}(A) is the spin s representation of SU(2). Using the formalism of second quantization, one can make a theory of fermions by requiring

[Ψ(x⃗, M), Ψ(y⃗, N)]_{+} = 0
[Ψ(x⃗, M), Ψ^{∗}(y⃗, N)]_{+} = δ_{MN}δ(x⃗ − y⃗)  (2)

or, alternatively, a theory of bosons by requiring

[Ψ(x⃗, M), Ψ(y⃗, N)] = 0
[Ψ(x⃗, M), Ψ^{∗}(y⃗, N)] = δ_{MN}δ(x⃗ − y⃗).  (3)

∗Mathematics Subject Classifications: 81705.

Key words: Spin-statistics theorem.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University and University of North Texas.

Published July 12, 2000.

The Fock space construction required to realize (1) and (2) or (1) and (3) are easy generalizations of what one finds for spin \frac{1}{2} in the text books of condensed matter physics. (See, for example, Fetter and Walecka, Quantum Theory of Many Particle Systems, McGraw-Hill, New York 1971.) It is clear here that spin does not determine statistics and that any positive response to Neuenschwander’s question must make assumptions that go beyond Euclidean invariance. For example, in 1940 Pauli assumed Lorentz invariance and also that he was dealing with a quantum field theory of non-interacting particles. Under these hypotheses, he proved the Spin-Statistics Theorem.

It took more than twenty years before the extension of Pauli’s result to interacting fields saw the light of day in the work of Lüders and Zumino and of Burgoyne. The extension exploited then recent developments in the general theory of quantized fields. This brings me to my second point which is a response to the question: How does invariance under Lorentz transformations bind statistics to spin, and, in particular, what properties of the Lorentz group are essential?

It is here that I advocate ignoring Henry David Thoreau’s advice (Walden Chapter II) “Simplify, simplify!” Instead:

complexify, complexify!

I illustrate the application of this slogan with Lorentz transformations. They are defined as 4 × 4 matrices satisfying

Λ^{T} GΛ = G

where

G = \{ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} \} .

For the direct application to physics, Λ is real, but one can also consider complex Λ satisfying the definition. For both cases (det Λ)^{2} = 1, so det λ = ±1. If det Λ = +1, we call Λ a proper Lorentz transformation; if det Λ = −1, improper. For real Lorentz transformations, there is another distinction according to the sign of Λ_{o}^{o}, sgn Λ^{o} _{o}. If sgn Λ_{o}^{o} = +1, we say that Λ is orthochronous; if sgn Λ^{o} _{o} = −1, that Λ inverts time.

The real Lorentz group is composed of four components, L_{+}^{↑}, L_{+}^{↓}, L^{↑}, and L^{↓}, disconnected from each other: The arrow superscript points up for orthochronous transformations and down for those inverting time. The subscript is + for proper transformations and - for improper. The particular Lorentz transformations I_{s}, I_{t}, and I_{st} are defined as follows:

Space inversion I_{s} : {x^{o}, x⃗} → {x^{o}, −x⃗}
Time inversion I_{t} : {x^{o}, x⃗} → {−x^{o}, x⃗}
Space-time inversion I_{st} : {x^{o}, x⃗} → {−x^{o}, −x⃗}

On the other hand, the Complex Lorentz group, which is denoted L(C), has two connected components, the proper complex Lorentz transformations L_{+}(C) and the improper L (C). One can pass from L_{+}^{↑} to L_{+}^{↓} through L_{+}(C). In particular the curve of complex Lorentz transformations

Λ_{t} = \{ \begin{matrix}cosh it & 0 & 0 & sinh it \\ 0 & cost & − sin t & 0 \\ 0 & sin t & cos t & 0 \\ sinh it & 0 & 0 & cosh it \end{matrix} \} , 0 ≤ t ≤ π

connects 1= Λ_{o} to -1= I_{st} = Λ_{π}. Similarly one can pass from L^{↑} to L^{↓} via L (C). The existence of these connections is indicated by the diagonal arrows in the diagram. How can this phenomenon possibly have anything to do with physics?

The answer is that in the general theory of quantized fields, there is a sequence of functions (or better, generalized functions) associated with the fields, the vacuum expectation values of monomials in the fields. For example, to a scalar field ϕ, there are associated the vacuum expectation values

F_{n}(x_{1}, … x_{n}) = (Ψ_{o}, ϕ(x_{1}) … ϕ(x_{n})Ψ_{o}).

If we take the transformation law of the states under the restricted Poincaré group as the unitary representation: {a, Λ} → U(a, Λ) (a is a translation of space-time, Λ a Lorentz transformation in L_{+}^{↑}), then the scalar field has a transformation law

U(a, Λ)ϕ(x)U(a, Λ)^{−1} = ϕ(Λx + a),

and in view of the invariance

U(a, Λ)Ψ_{o} = Ψ_{o}

of the vacuum state Ψ_{o}, we have

F_{n}(Λx_{1} + a, Λx_{2} + a, … Λx_{n} + a) = F_{n}(x_{1}, … x_{n}) .

This relation implies that F_{n} depends only on the difference variables

ξ_{1} = x_{2} − x_{1}, ξ_{2} = x_{3} − x_{2}, …, ξ_{n−1} = x_{n} − x_{n−1}

and so, in a change of notation, we write

F_{n}(Λξ_{1}, … Λξ_{n−1}) = F_{n}(ξ_{1}, … ξ_{n−1}).

The next step in this line of argument is to recognize that the spectral condition, i.e., the physical requirement that the energy momentum vector of the state of the system lies in or on the cone \overline{V}_{+}, implies that F_{n} is the Fourier transform of a distribution G_{n} which vanishes unless each of its n−1 arguments lies in \overline{V}_{+}. In formulae,

F_{n}(ξ_{1}, … ξ_{n−1}) = ∫ d^{4}p, … d^{4}p_{n−1} exp i ∑_{j=1}^{n−1} p_{j} · ξ_{j} G_{n}(p_{1}, … p_{n−1}).

Here G_{n}(p_{1}, … p_{n−1}) = 0 unless p_{1}, … p_{n−1} all lie in \overline{V}_{+}, where

\overline{V}_{+} = \{p; p · p = (p^{o})^{2} − p⃗^{2} ≥ 0, p^{o} ≥ 0\}.

Because of the support properties of G_{n}, this formula can be extended from a Fourier transform to a Laplace transform,

F_{n}(ξ_{1} + iη_{1}, … ξ_{n−1} + iη_{n−1})
= ∫ d^{4}p_{1} … d^{4}p_{n−1} exp i ∑_{j=1}^{n−1} p_{j} · (ξ_{j} + iη_{j})G_{n}(p_{1}, … p_{n−1}),

and so extended defines an analytic function for all ξ_{1} + iη_{1}, … ξ_{n−1} + iη_{n−1} lying in the tube

ξ_{j} ∈ ℝ^{4}, j = 1, … n − 1 and η_{j} ∈ V_{+}, j = 1, … n − 1

(V_{+} is the interior of \overline{V}_{+}). So extended, F_{n} remains Lorentz invariant:

F_{n}(Λζ_{1}, Λζ_{2}, … Λζ_{n−1}) = F_{n}(ζ_{1}, … ζ_{n−1}) for all Λ ∈ L_{+}^{↑}  (4)

where ζ_{1} = ξ_{1} + iη_{1}, … ζ_{n−1} = ξ_{n−1} + iη_{n−1} is any point of the tube.

For a fixed point ζ_{1} … ζ_{n−1} of the tube, the left-hand side of this equation (4) is an analytic function of Λ. This suggests that F_{n} can be extended by analytic continuation to points of the form Λζ_{1}, … Λζ_{n−1}, where Λ is an arbitrary proper complex Lorentz transformation; the set of all such points form the extended tube. The BHW Theorem (1957) says that, so extended, F_{n} is analytic and single-valued in its argument ζ_{1} … ζ_{n−1} and is invariant under L_{+}(C), i.e., equation (4) is valid for all Λ ∈ L_{+}(C).

It is an important and perhaps somewhat surprising property of the extended tube that it contains real points; they are called Jost points because R. Jost first characterized them as follows.

Theorem ξ_{1}, … ξ_{n−1} is a real point of the extended tube if and only if for every sequence λ_{1}, … λ_{n−1} of non-negative real numbers satisfying ∑_{j=1}^{n−1} λ_{j} > 0, ∑_{j=1}^{n−1} λ_{j}ξ_{j} is space-like.

An immediate corollary of this theorem of Jost is the existence of an open set in the real vector variables x_{1}, … x_{n} wherein the vacuum expectation value (Ψ_{o}, ϕ(x_{1}) … ϕ(x_{n})Ψ_{o}) is analytic in the x_{1} … x_{n}. Furthermore, at such Jost points, application of (4) and its analogues for vacuum expectation values of monomials in the components of tensor or spinor fields or their adjoints yield identities at the core of the “modern” proofs of CPT Symmetry (Jost 1957) and the Spin-Statistics Theorem (Luders and Zumino 1958, Burgoyne 1958). I want to emphasize that no assumption of invariance under I_{st} = − 1 has been made in these arguments.

My third remark involves a little more detail from Burgoyne’s proof, substantiating the remarks I have just made. I treat first the case of a scalar field, ϕ, for which Burgoyne wants to show that

[ϕ(x), ϕ^{∗}(y)]_{+} = 0 for (x − y)^{2} < 0

leads to the conclusion that ϕ is zero. Take the vacuum expectation value of this relation to obtain

(Ψ_{o}, ϕ(x)ϕ^{∗}(y)Ψ_{o}) + (Ψ_{o}, ϕ^{∗}(y)ϕ(x)Ψ_{o}) = 0 for (x − y)^{2} < 0.  (5)

Each of the two terms is the boundary value of a function analytic in the tube:

(Ψ_{o}, ϕ(x)ϕ^{∗}(y)Ψ_{o}) = lim_{η∈V_{+}^{η→0}} F_{ϕϕ^{∗}} (y − x + iη)
(Ψ_{o}, ϕ^{∗}(y)ϕ(x)Ψ_{o}) = lim_{η∈V_{+}^{η→0}} F_{ϕ^{∗}ϕ} (x − y + iη)

Both F_{ϕϕ^{∗}} and F_{ϕ^{∗}ϕ} are invariant under L_{+}^{↑} and therefore by the previous argument analytic in the extended tube and invariant under L_{+}(C). The real points of the extended tube are here all pairs x, y for which x − y is space-like, so the relation (5) implies that

F_{ϕϕ^{∗}}(ζ) + F_{ϕ^{∗}ϕ}(−ζ) = 0  (6)

throughout the extended tube, or, using the invariance of F_{ϕ^{∗}ϕ} under Λ = − 1 = I_{st},

F_{ϕϕ^{∗}}(ζ) + F_{ϕ^{∗}ϕ}(ζ) = 0.  (7)

If we pass to the limit η → 0 with η ∈ V_{+} in this relation, we have for all x and y the relation

(Ψ_{o}, ϕ(x)ϕ^{∗}(y)Ψ_{o}) + (Ψ_{o}, ϕ^{∗}(−y)ϕ(−x)Ψ_{o}) = 0

between distributions. Smearing in x with the test function f(x) and in y with \overline{f(y)}, we get for the first term ||ϕ(f)^{∗}Ψ_{o}||^{2} and for the second ||ϕ(f̂)Ψ_{o}||^{2} = 0 for all test functions f. Here f̂(x) = f(−x). So ϕ(f)^{∗} and ϕ(f) both annihilate the vacuum. To conclude from these results that ϕ and ϕ^{∗} are actually zero requires some knowledge of the relations of ϕ and ϕ^{∗} with the rest of the fields of the field theory; I will not say more about that. However, I will add that the extension of the above discussion to treat the case of arbitrary spinor and tensor fields basically only involves the replacement of (4), which expresses invariance under L_{+}(C), by a transformation law involving the representation which expresses the nature of the tensors or spinors under the action of L_{+}(C). Explicitly, for ϕ a component of a tensor field, the argument goes just as in the scalar case, because the tensor transformation law reduces to (6) and (7). On the other hand, for ϕ a component of a spinor field, Burgoyne undertakes to show that the vanishing of the commutator of ϕ(x) and ϕ^{∗}(y) for space-like (x − y) implies

∥ϕ(f)^{∗}Ψ_{o}∥^{2} = ∥ϕ(f̂)Ψ_{o}∥^{2} = 0.

He has to overcome the apparent difficulty that here (6) is replaced by

F_{ϕϕ^{∗}}(ζ) − F_{ϕ^{∗}ϕ}(−ζ) = 0

throughout the extended tube. But now the spinor character of ϕ implies that

F_{ϕ^{∗}ψ}(−ζ) = −F_{ϕ^{∗}ϕ}(ζ).

The minus signs compensate and the rest of the proof goes the same way as for a scalar or tensor field.

Finally, I should mention that every result that I have talked about presupposes a space of states that is a Hilbert space with positive metric; Faddeev-Popov ghosts are thereby excluded; they violate the spin-statistics connection.

Discussion

E. Wichmann: How is your assertion about the non-existence of a spinstatistics connection in Euclidean invariant theory related to the paper of Michael Berry (Proc. Roy. Soc. Lond. A 453 (1997) 1771-1790) in which under some assumptions he outlined a proof. What are those assumptions?

A. Wightman: I have not seen the paper of Michael Berry. (with some delay) I want to thank Dave Jackson for providing me with a photocopy of Berry’s paper and the organizers for accepting a delayed response to Eyvind’s question.

Berry bases his proof on the assumption of the existence of what he calls a transportable spin-basis. He constructs such a basis for the special case of two particles, N = 2, but leaves open the case of general N.

A. S. Wightman Department of Mathematics, Princeton University Fine Hall, Washington Rd Princeton, NJ 08544-0001 USA email: wightman@princeton.edu