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\markboth{The spin-statistics connection}{ A. S. Wightman}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Mathematical Physics and Quantum Field Theory, \newline
Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 207--213\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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The spin-statistics connection: \\ Some pedagogical
remarks in response to Neuenschwander's question
\thanks{ {\em Mathematics Subject Classifications:} 81705.
\hfil\break\indent
{\em Key words:} Spin-statistics theorem.
\hfil\break\indent
\copyright 2000 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Published July 12, 2000.} }
\date{}
\author{ A. S. Wightman }
\maketitle
In 1994, D.E. Neuenschwander posed a question (American Journal of
Physics {\bf 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 $\Psi (\vec{x}, M), M= -s, -s+1, \ldots (s-1), s,$
satisfying
\begin{equation}
V(\vec{a}, A) \Psi (x, M) V (\vec{a}, A)^{-1} = \sum_{L}
{\mathcal{D}}^{(s)} (A^{-1})_{ML} \Psi (R(A) \vec{x} + \vec{a}, L)
\label{apples}
\end{equation}
where $\{\vec{a}, A\} \rightarrow V(\vec{a}, A)$ 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 $A
\rightarrow {\mathcal{D}}^{(s)}(A)$ is the spin $s$ representation of
$SU(2)$. Using the formalism of second quantization, one can make a
theory of fermions by requiring
\begin{eqnarray}
&\left[ \Psi (\vec{x}, M), \Psi (\vec{y}, N) \right]_+ = 0& \nonumber \\
&\left[ \Psi (\vec{x}, M), \Psi^{\ast} (\vec{y}, N) \right]_+ =
\delta_{MN} \delta (\vec{x} - \vec{y})&
\label{oranges}
\end{eqnarray}
or, alternatively, a theory of bosons by requiring
\begin{eqnarray}
\left[ \Psi (\vec{x}, M), \Psi (\vec{y}, N) \right]_{\_} &=& 0 \nonumber \\
\left[ \Psi (\vec{x}, M), \Psi^{\ast} (\vec{y}, N) \right]_{\_} &=&
\delta_{MN} \delta (\vec{x} - \vec{y}).
\label{pears}
\end{eqnarray}
The Fock space construction required to realize (\ref{apples}) and
(\ref{oranges}) or (\ref{apples}) and (\ref{pears}) 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, {\it 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 L\"uders 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:
\smallskip
\centerline{complexify, complexify!}
\smallskip
I illustrate the application of this slogan with Lorentz
transformations. They are defined as $4 \times 4$ matrices satisfying
$$
\Lambda^T G \Lambda = G
$$
where
\[ G = \left\{ \begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{array}
\right\}. \]
For the direct application to physics, $\Lambda$ is real, but one can
also consider complex $\Lambda$ satisfying the definition. For both
cases $(\det \Lambda)^2 = 1$, so $\det \lambda = \pm 1$. If $\det
\Lambda = +1$, we call $\Lambda$ a {\it proper} Lorentz
transformation; if $\det \Lambda = -1$, {\it improper}. For real
Lorentz transformations, there is another distinction according to the
sign of $\Lambda^o_o$, sgn $\Lambda^o \, _o$. If sgn $\Lambda^o_o =
+1$, we say that $\Lambda$ is {\it orthochronous}; if sgn
$\Lambda^o \, _o = -1$, that $\Lambda$ inverts time.
The real Lorentz group is composed of four components,
$L_+^{\uparrow}$, $L_+^{\downarrow}$, $L_{\_}^{\uparrow}$,
and $L_{\_}^{\downarrow}$, disconnected from each other:
\begin{figure}
\begin{center}
\epsfig{file=fig1.eps, scale=0.60} %
\end{center}
\end{figure}
\noindent 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:
\begin{center}
Space inversion $I_s: \{ x^o, \vec{x} \} \rightarrow \{ x^o, -\vec{x}
\}$\\
Time inversion $I_t: \{ x^o, \vec{x} \} \rightarrow \{ -x^o, \vec{x}
\}$\\
Space-time inversion $I_{st}: \{ x^o, \vec{x} \} \rightarrow \{ -x^o, -\vec{x}
\}$\\
\end{center}
\bigskip
On the other hand, the Complex Lorentz group, which is denoted
$L$({\bf C}), has two connected components, the proper complex Lorentz
transformations $L_+$({\bf C}) and the improper $L_{\_}$({\bf C}). One
can pass from $L_+^{\uparrow}$ to $L_+^{\downarrow}$ through
$L_+$({\bf C}). In particular the curve of complex Lorentz
transformations
\[ \Lambda_t = \left\{ \begin{array}{cccc}
\cosh it & 0 & 0 & \sinh it\\
0 & cost & -\sin t & 0 \\
0 & \sin t & \cos t & 0 \\
\sinh it & 0 & 0 & \cosh it
\end{array}
\right\} ~~ , \quad\quad 0 \leq t \leq \pi
\]
connects {\bf 1}$ = \Lambda_o$ to -{\bf 1}$ = I_{st} =
\Lambda_{\pi}$. Similarly one can pass from $L_{\_}^{\uparrow}$ to
$L_{\_}^{\downarrow}$ via $L_{\_}$({\bf 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 $\phi$, there are associated the
vacuum expectation values
$$
F_n (x_1, \ldots x_n) = (\Psi_o, \phi (x_1) \ldots \phi (x_n) \Psi_o).
$$
If we take the transformation law of the states under the restricted
Poincar\'e group as the unitary representation: $\{ a, \Lambda \}
\rightarrow U (a, \Lambda)$ ($a$ is a translation of space-time,
$\Lambda$ a Lorentz transformation in $L_+^{\uparrow}$), then the
scalar field has a transformation law
$$
U(a, \Lambda) \phi (x) U (a, \Lambda)^{-1} = \phi (\Lambda x + a),
$$
and in view of the invariance
$$
U(a, \Lambda ) \Psi_o = \Psi_o
$$
of the vacuum state $\Psi_o$, we have
$$
F_n (\Lambda x_1 + a, \Lambda x_2 + a, \ldots \Lambda x_n + a ) = F_n
(x_1, \ldots x_n) ~ .
$$
This relation implies that $F_n$ depends only on the difference
variables
$$
\xi_1 = x_2 - x_1, ~ \xi_2 = x_3 - x_2, ~ \ldots, ~ \xi_{n-1} = x_n - x_{n-1}
$$
and so, in a change of notation, we write
$$
F_n (\Lambda \xi_1, \ldots \Lambda \xi_{n-1} ) = F_n (\xi_1, \ldots
\xi_{n-1}).
$$
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,
$$
F_n (\xi_1, \ldots \xi_{n-1} ) = \int d^4 p, \ldots d^4 p_{n-1} \exp i
\sum_{j=1}^{n-1} ~ p_j \cdot \xi_j ~ G_n (p_1, \ldots p_{n-1}).
$$
Here $G_n (p_1, \ldots p_{n-1}) = 0$ unless $p_1, \ldots p_{n-1}$ all
lie in $\overline{V}_+$, where
$$
\overline{V}_+ = \{ p ; p \cdot p = (p^o)^2 - \left. {\vec{p}}\right.^2 \geq 0, ~ p^o
\geq 0 \}.
$$
Because of the support properties of $G_n$, this formula can be
extended from a Fourier transform to a Laplace transform,
\begin{eqnarray*}
\lefteqn{ F_n ( \xi_1 + i \eta_1, \ldots \xi_{n-1} + i \eta_{n-1} )}\\
&= & \int d^4
p_1 \ldots d^4 p_{n-1}
\exp i \sum_{j=1}^{n-1} p_j \cdot ( \xi_j + i \eta_j)
G_n(p_1, \ldots p_{n-1}),
\end{eqnarray*}
and so extended defines an analytic function for all $\xi_1 + i
\eta_1, \ldots \xi_{n-1} + i \eta_{n-1}$ lying in the tube
$$
\xi_j \in {\mathbb{R}}^4,~ j=1, \ldots n-1 ~~ {\rm{and}} ~~ \eta_j \in
V_+ , ~~ j=1, \ldots n-1
$$
$(V_+$ is the interior of $\overline{V}_+$). So extended, $F_n$ remains
Lorentz invariant:
\begin{equation}
F_n (\Lambda \zeta_1, \Lambda \zeta_2, \ldots \Lambda \zeta_{n-1} ) = F_n
(\zeta_1, \ldots \zeta_{n-1})~~ {\rm{for~~all}}~~ \Lambda \in L_+^{\uparrow}
\end{equation}
where $\zeta_1 = \xi_1 + i \eta_1 , \ldots \zeta_{n-1} = \xi_{n-1} + i
\eta_{n-1}$ is any point of the tube.
For a fixed point $\zeta_1 \ldots \zeta_{n-1}$ of the tube, the left-hand side
of this equation (4) is an analytic function of $\Lambda$. This
suggests that $F_n$ can be extended by analytic continuation to points
of the form $\Lambda \zeta_1 , \ldots \Lambda \zeta_{n-1}$, where $\Lambda$
is an arbitrary proper complex Lorentz transformation; the set of all
such points form the {\it extended tube}. The BHW Theorem (1957)
says that, so extended, $F_n$ is analytic and single-valued in its
argument $\zeta_1 \ldots \zeta_{n-1}$ and is invariant under $L_+$({\bf C}),
i.e., equation (4) is valid for all $\Lambda \in L_+$({\bf 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.
\paragraph{Theorem}
$\xi_1, \ldots \xi_{n-1}$ is a real point of the extended tube if
and only if for every sequence $\lambda_1, \ldots \lambda_{n-1}$ of
non-negative real numbers satisfying
$\sum_{j=1}^{n-1} \lambda_j > 0$,
$\sum_{j=1}^{n-1} \lambda_j \xi_j$
is space-like. \medskip
An immediate corollary of this theorem of Jost is the existence of an
open set in the real vector variables $x_1, \ldots x_n$ wherein the
vacuum expectation value $(\Psi_o, \phi (x_1) \ldots \phi (x_n)
\Psi_o)$ is analytic in the $x_1 \ldots x_n$. 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 $I_{st} = -$ {\bf 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, $\phi$, for which Burgoyne wants to show that
$$
\left[ \phi (x), \phi^{\ast} (y) \right]_+ = 0 ~~{\rm{for}}~~ (x-y)^2 <
0
$$
leads to the conclusion that $\phi$ is zero. Take the vacuum
expectation value of this relation to obtain
\begin{equation}
(\Psi_o, \phi (x) \phi^{\ast} (y) \Psi_o) + (\Psi_o, \phi^{\ast} (y)
\phi (x) \Psi_o ) = 0 \quad\mbox{for } (x-y)^2 < 0 .
\end{equation}
Each of the two terms is the boundary value of a function analytic in
the {\it tube}:
\begin{eqnarray*}
(\Psi_o, \phi (x) \phi^{\ast} (y) \Psi_o ) & = & \lim_{\buildrel ~\eta
\rightarrow 0 \over {~\eta \in V_+}} ~~ F_{\phi \phi^{\ast}} ~(y - x
+ i \eta)\\
(\Psi_o, \phi^{\ast} (y) \phi (x) \Psi_o ) &=& \lim_{\buildrel ~\eta
\rightarrow 0 \over {~\eta \in V_+}} ~~ F_{\phi^{\ast} \phi} ~(x - y + i \eta)\\
\end{eqnarray*}
Both $F_{\phi \phi^{\ast}}$ and $F_{\phi^{\ast}\phi}$ are invariant
under $L_+^{\uparrow}$ and therefore by the previous argument analytic
in the extended tube and invariant under $L_+$({\bf 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
\begin{equation}
F_{\phi \phi^{\ast}} (\zeta) + F_{\phi^{\ast}\phi} (-\zeta) = 0
\end{equation}
throughout the extended tube, or, using the invariance of
$F_{\phi^{\ast}\phi}$ under $\Lambda = -$ {\bf 1} = $I_{st}$,
\begin{equation}
F_{\phi \phi^{\ast}} (\zeta) + F_{\phi^{\ast}\phi} (\zeta) = 0.
\end{equation}
If we pass to the limit $\eta \rightarrow 0$ with $\eta \in V_+$ in this
relation, we have for all $x$ and $y$ the relation
$$
(\Psi_o, \phi(x) \phi^{\ast} (y) \Psi_o) + (\Psi_o, \phi^{\ast} (-y)
\phi (-x) \Psi_o) = 0
$$
between distributions.
Smearing in $x$ with the test function $f(x)$ and in $y$ with
$\overline{f(y)}$, we get for the first term $ || \phi (f)^{\ast}
\Psi_o ||^2$ and for the second $|| \phi (\hat{f}) \Psi_o ||^2 = 0$ for all
test functions $f$.
Here $\hat{f} (x) = f (-x)$. So $\phi(f)^{\ast}$ and $\phi(f)$ both
annihilate the vacuum.
To conclude from these results that $\phi$ and
$\phi^{\ast}$ are actually zero requires some knowledge of the
relations of $\phi$ and $\phi^{\ast}$ 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 $L_+$({\bf
C}), by a transformation law involving the representation which
expresses the nature of the tensors or spinors under the action of
$L_+$({\bf C}).
Explicitly, for $\phi$ 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 $\phi$ a component of
a spinor field, Burgoyne undertakes to show that the vanishing of the
commutator of $\phi (x)$ and $\phi^{\ast}(y)$ for space-like $(x-y)$
implies
$$
\Vert \phi(f)^{\ast} \Psi_o \Vert^2 ~=~ \Vert \phi (\hat{f}) \Psi_o
\Vert^2 = 0.
$$
He has to overcome the apparent difficulty that here (6) is replaced
by
$$
F_{\phi \phi^{\ast}} (\zeta) - F_{\phi^{\ast}\phi} (-\zeta) = 0
$$
throughout the extended tube. But now the spinor character of $\phi$
implies that
$$
F_{\phi^{\ast}\psi} (-\zeta) = -F_{\phi^{\ast}\phi} (\zeta).
$$
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.
\bigskip
\noindent Discussion
E. Wichmann: How is your assertion about the non-existence of a
spin-statistics connection in Euclidean invariant theory related to
the paper of Michael Berry (Proc. Roy. Soc. Lond. A {\bf 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 $N$.
\bigskip
\noindent{\sc A. S. Wightman}\\
Department of Mathematics, Princeton University\\
Fine Hall, Washington Rd\\
Princeton, NJ 08544-0001 USA \\
email: wightman@princeton.edu
\end{document}