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.
Another version of the draft (moved from page spin and statistics)
1. History
In 1920? S. K. Bose submitted a paper (1) to the Royal Society, in which he claimed that the Planck distribution of light (2) can be derived from statistical mechanics of the photons postulated by Einstein (3). The paper was rejected, on the grounds that it contained nothing new. He then sent it to Einstein, with a plea to get it published; he knew that it did contain some new idea. Einstein saw perhaps more that Bose; he recommended it to Zeitschrift. In fact, Bose had applied statistical dynamics to the system of fields that arise, namely, the electric and magnetic fields which make up the classical phase space. Einstein then published his version, (4), in which the degrees of freedom or particles, photons, were treated, but a new way of counting the different points of phase-space was used: the particles lose their identity, and assemblies of photons obey a different statistical law from classical particles. This new law is now called Bose-Einstein statistics. For example, consider a point in the phase-space of two particles, the first being placed at x with momentum p and the second being placed at y with momentum q. This gives us the point (x,p),(y,q). In what is now called classical phase-space, this is a different point from (y,q),(x,p) unless x=y and p=q. In statistical mechanics, each point of phase space is regarded as being equally likely. The question then arises, how do we count the number of points giving rise to a specified electromagnetic field which arises from a certain distribution of photons? Bose had assumed that one should count the number of different field configurations, not noticing that this does not lead to the same distribution as assuming that actual photons were present. Einstein added the assumption that as photons are indistinguishable, one should include only once all configurations which are related by a permutation of their classical phase points. This differs from a classical view of the particles, and the photon is thus a Bose-Einstein particle. Particles with this property are now called bosons.
Before the arrival of wave-mechanics, the Bohr model of the atom (5) envisaged that the electrons of the atom were placed in orbits around the nucleus, which was of charge \(Z\ ,\) the atomic number. Each allowed orbit was of a radius that required that the electron's angular momentum was an integer times a quantum unit of angular momentum. After Goudschmidt and Uhlenbeck had suggested that the electron had an intrinsic spin (6), Pauli invented the Pauli matrices (7) which permitted the electron to have a spin \(\pm 1/2\) in say the \(z\)-direction. The state of an electron in an atom is labelled by two parameters, its angular momentum, an integer, and its spin, \(\pm 1/2\ .\) One difficulty with the atomic model was then found by Pauli: only two of the allowed configurations of electrons for each orbit can be occupied (8). This rule was called "Pauli's exclusion principle", meaning that an allowed orbit might be empty, or might be occupied by one or two electrons, but not more. When occupied by two electrons, their spins must be opposite: the same state cannot be occupied by more than one electron.
De Broglie suggested that matter, such as electrons, might exhibit wavelike properties (9), just as electromagnetic waves exhibit particle-like properties. This was observed by Davisson and Germer {10}. Heisenberg (11) invented the "new" quantum mechanics in 1925; an observable in the new theory is represented by a matrix, rather than by a number. Schrödinger wrote down (12) his equation to express the idea that matter is a wave: any wave must obey a wave-equation. He tried to write down a relativistic equation, but this led to what was later called the Klein-Gordon equation, after their papers (13,14), and he could not work out what the scalar product should be. This work was not published. Schrödinger spent the Summer of 1926 laying out the full theory, including a theory of several particles. Observables were represented by differential operators, instead of matrices, and the dynamics appeared as a time-dependence of the wave-function, rather than as a time-dependence of the operators. The Pauli exclusion principle could be included if one limited the wave-function of several particles to be anti-symmetric under an odd permutation of the position variables of the particles; this was for particles of spin zero. For particles of spin \(1/2\ ,\) the Pauli exclusion principle can be incorporated if the wave-function changes sign under an odd permutation of the positions and the values of the spins.
Dirac argued (15) that solutions to the Schrödinger equation led to the same physics as Heisenberg's new theory. Jordan and Wigner second-quantized the Schrödinger equation (16). In place of the dynamical variables, position and momentum, they used creation and annihilation operators, \(a^*(k)\) and \(a(k)\ ,\) where \(k\) is the momentum of the particle. For bosons, they derived the commutation relations
\( [a(k),a*{p}]:=a(k)a^*(p)-a^*(p)a(k)=\delta(k-p)\) (1)
and for particles obeying the Pauli principle, with anti-symmetric wave functions, they used
\( \{b(k),b^*(p)\}:=b(k)b^*(p)+b^*(p)a(k)=\delta(k-p)\) (2)
In these equations, \(\delta\) is the Dirac distribution, and \(p, k\) are three-vectors representing the momenta of the particles.
Fermi studied the statistical consequences of the second-quantized Schrödinger equation (16) in the early thirties.
Dirac solved the problem of constructing a relativistic model (17) by limiting his equation to a first-order partial differential equation, but with several components, which gave the electron a spin of \(1/2\ .\) For some years it was believed by many physicists that the scalar equation, describing a particle of spin zero, did not allow a positive-definite scalar product to be Lorentz invariant; indeed, there is no local scalar product, but there is a non-local expression which is Lorentz invariant. By the mid-thirties, the relativistic theories of particles of zero spin and spin 1 were treated as bosons, using eq.~{1); local quantum field theories of spin 1/2 and 3/2 used eq.~(2). The latter particles were said to obey Fermi-Dirac statistics, and are now called fermions. The anti-commutation relations assumed for the creation and annihilation operators of particles of half-odd-integer spin not only guaranteed that these particles obeyed the Pauli exclusion principle, but also made sure that fields thought to be observable, such as the density-current field \(\bar{\psi}\gamma_\mu\psi\ ,\) would commute at space-like separations, which is needed for observable quantities in a relativistic theory.
In 1939 Fierz produced a paper, whose title, translated into English, is "On the relativistic theory of free particles with arbitrary spin" (18); this assumed that the usual connection between spin and statistics held. In 1940, Pauli (19) showed that for free fields, one cannot make the wrong choice: if we postulate eq.~(2) for a particle of integer spin then the field must be zero; also we get zero if we postulate eq.~(1} for a field with half-odd-integer spin. We shall give a proof of a similar result for any interacting relativistic quantum field theory that satisfies the Wightman axioms (20). While this was a good result in its time, it fails to deal with the following questions.
1) To show that the wrong connection leads to a contradiction does not mean that the right connection is the only choice.
2) It omits any model with parastatistics, which represents the permutation group by a matrix, not just \(\pm 1\ .\)
3) It omits fields with infinitely many components, for which the wrong connection is possible.
More recently, Haag, Doplicher and Roberts have addressed (21) the statistics that can arise from a theory based on Haag's idea that local observables must commute at space-like separation (22). They find that excitations of half-odd-integer spin must be represented using parafermion operators, and that excitations of integer spin must use paraboson operators. The case (3) is ruled out by an axiom limiting the dimensionality of the local algebras, called the "separation axiom".
We refer you to the work of Haag, Doplicher and Roberts; here we limit our proof to Wightman theory.
2. The proof of the connection between spin and statistics
We start by showing that if two Wightman fields \(\psi\) and \(\varphi\) commute at space-like separation, and \(\psi^*\) and \(\varphi\) anti-commute at space-like separation, then one or the other must be zero. This was first proved by Dell'Antonio (23).
It is enough to show that if \([\psi(x),\varphi(y)]=0\) and \(\{\psi^*(x),\varphi(y)\}=0\) whenever \(x\) and \(y\) are space-like separated, then either \(\psi=0\) or \(\varphi=0\ .\) To prove this, let \(f,g\) be test-functions of space and time with compact support. Then we have
\(\langle\Psi_0,\varphi((f)^*\psi(g)^*\psi(g)\varphi(f)\psi_0\rangle=\|\psi(g)\varphi(f)\Psi_0\|^2\geq 0\ .\)
If the supports of \(f\) and \(g\) are spacelike separated, then the left-hand side of this inequality is equal to
\(-\langle\Psi_0,\psi(g)^*\psi(g)\varphi(f)^*\varphi(f)\Psi_0\rangle\ .\)
In any Wightman theory there is a unique vacuum, and this implies the cluster decomposition theorem (24). Let us translate the function \(g\) to infinity in a space-like direction. Then by the cluster decomposition theorem, the above expression converges to
\(-\langle\Psi_0,\psi(g)^*\psi(g)\Psi_0\rangle\langle\Psi_0,\varphi(f)^*\varphi(f)\Psi_0\rangle=-\|\psi(g)\Psi_0\|^2\,\|\varphi(f)\Psi_0\|^2\)
and this is not positive. Comparing with the above, we see that the only consistent value for the limit is zero, so that either \(\psi(g)\Psi_0=0\) or \(\varphi(f)\Psi_0=0\ .\) The Reeh-Schlieder theorem implies that the vacuum state is both cyclic and separating for any local operator (24). Separating means that \(\psi(g)\Psi_0=0\) implies that \(\psi(g)=0\ .\) So if \(\psi\neq 0\ ,\) there exists a test-function of compact support, \(g\ ,\) such that \(\psi(g)\Psi_0\neq 0\ .\) But then \(\varphi(f)\Psi_0=0\ ,\) which by Reeh-Schlieder shows that \(\varphi(f)=0\ ;\) this then holds for all \(f\) of compact support, which shows that \(\varphi=0\ ,\) as claimed.
We now turn to the spin-statistics theorem proper. We shall need to use the analytic properties of the Wightman functions; in particular that the "two-point function" \(W(x-y):=\langle\Psi_0,\varphi(x)\varphi^*(y)\Psi_0\rangle\) is a distribution in the four-vector \(x-y\ ,\) and has an analytic continuation in \(x-y\) in the forward tube and the backward tube. It is also analytic at real points where \(x-y\) is spacelike. Here, \(x\) and \(y\) are complex four-vectors, and the forward tube of a complex four-vector \(z\) is the set of points where \({\rm Im}z\) lies in the forward light-cone. The backward tube is defined as the set of points \(z\) where \(-z\) lies in the forward tube. The wrong connection between spin and statistics says that if \(x-y\) is a space-like vector, then
\(\{\psi(x),\psi^*(y)\}=0\) if \(\psi\) has integer spin, or
\([\psi(x),\psi^*(y)]=0\) if \(\psi\) has half-odd-integer spin.
THEOREM Let \(\psi\)n be an irreducible spinor field in a Wightman theory. Then the wrong connection of spin with statistics implies that \(\psi=0.\)
PROOF. The Wightman distributions are the boundary values of holomorphic functions \(W\) and \(\hat{W}\ ,\) thus\[\langle\Psi_0,\psi(x)\psi^*(y)\Psi_0\rangle=\lim_{\eta\rightarrow 0}W(x-y-i\eta)\ ,\] and
\(\langle\Psi_0,\psi^*(y)\psi(x)\Psi_0\rangle=\lim_{\eta\rightarrow 0}\hat{W}(y-x-i\eta)\ .\)
In these equations, \(\eta\) is a real four-vector lying in the forward cone. The wrong connection says that if \(x-y\) is spacelike, then
\(W(x-y)=\mp \hat{W}(y-x)\) (3).
with minus for integer-spin and plus for half-odd-integer spin. But spacelike vectors are in the domain of holomorphy of both functions, and form an open set of real dimension 4, so the relation (3) holds at all points of analyticity\[W(\zeta)=\mp \hat{W}(-\zeta)\] (4).
Now, the real Lorentz group, without reflections, is a symmetry group of the system, and the vacuum is a fixed point under this. It follows from the Hall-Wightman-Bargmann theorem that the vacuum expectation values, the Wightman functions, are invariant under the connected complex Lorentz group. The space-time reflection operator,\(\Lambda=-I\) is an element of this group, though it is not in the connected real Lorentz group. For a spinor field with \(j\) undotted indices, the transformation law under \(\Lambda=-I\) is to multiply the field by \((-1)^j\ .\) For the product of the field and its hermitian conjugate field, we get the factor \((-1)^J\ ,\) where \(J\) is the number of undotted indices in \(\psi\psi^*\ ,\) which is the total number of indices in \(\psi\ .\) This is even if the spin-value is integral, and is odd if the spin-value is half-odd-integral. Thus we get the relation
\(\hat{W}(\zeta)=(-1)^J\hat{W}(-\zeta)\)
for all points of analyticity \(\zeta\ .\) Combined with (2), this gives us the relation
\(W(\zeta)=-\hat{W}(\zeta)\) at all points of analyticity. We now pass to the limit as \(\eta\rightarrow 0\ ,\) to get the distribution relation
\(\langle\Psi_0,\psi(x)\psi^*(y)\Psi_0\rangle+\langle\Psi_0,\psi^*(-y)\psi(-x)\Psi_0\rangle=0\) (3)
Now let \(f\) be a test-function, and put \(\hat{f}(x)=f(-x), x\in R^4.\ .\) Note that \(\psi^*(f)=[\psi(\bar{f})]^*\ ,\) \(\psi(f)=\int dx f(x)\psi(x)\) and \(\psi(\hat{f})=\int dx \psi(x)f(-x)=\int dx \psi(-x)f(x)\ ,\) we get
\(\|\psi(f)\Psi_0\|^2+\|\psi(\hat{f})\Psi_0\|^2=\langle\Psi_0,\psi(f)\psi(f)^*\Psi_0\rangle+\langle\Psi_0,\psi(\hat{f})^*\psi(\hat{f})\Psi_0\rangle\)
\(=\int dx\,dy\, f(x)f(y)^*[\langle\Psi_0,\psi(x)\psi^*(y)\Psi_0\rangle+\langle\Psi_0, \psi*(-y)\psi(-x)\Psi_0\rangle]=0\ .\)
Thus, for all test-functions \(f\ ,\) \(\psi(f)\Psi_0=0\ ,\) so we conclude that the field \(\psi\) vanishes, as in the first part of this section.
REFERENCES
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