Raymond Frederick Streater
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I. Introduction | I. Introduction | ||
| − | Einstein's theory 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 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 expressed this by requiring that the state on ''n'' photons should be symmetric under the permutation group <math>S_n</math>. | + | 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 <math>S_n</math>. |
| − | To get the (nearly) correct energy levels for the hydrogen atom from Heisenberg's new quantum mechanics, Pauli 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 | + | 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 <math>a_j^*</math> for a photon in the normalised state <math>j</math> and its hermitian conjugate, the annihilation operators <math>a_j</math>. The physicists postulated commutation relation, <math>a_ja_k^*-a_k^*a_j=\hbar i\delta_{jk}</math>, | operators <math>a_j^*</math> for a photon in the normalised state <math>j</math> and its hermitian conjugate, the annihilation operators <math>a_j</math>. The physicists postulated commutation relation, <math>a_ja_k^*-a_k^*a_j=\hbar i\delta_{jk}</math>, | ||
with all <math>a_j</math>, (and hence all <math>a_j^*</math>) commuting among themselves. Then they | with all <math>a_j</math>, (and hence all <math>a_j^*</math>) commuting among themselves. Then they | ||
| − | combined this with the evident existence of a ''no-particle state'', the vacuum, <math>\Psi_0</math> with the property <math>a_j\Psi_0=0</math>, 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 <math>\hbar</math>, and the exclusion principle can be satisfied if the space of <math>n</math> electrons is deemed to be totally ''anti-symmetric'' under the group <math>S_n</math>. Jordan and Wigner introduced creators and annihilators for electrons by operators <math>b_j^*,b_j</math> obeying | + | combined this with the evident existence of a ''no-particle state'', the vacuum, <math>\Psi_0</math> with the property <math>a_j\Psi_0=0</math>, 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 <math>\hbar</math>, and the exclusion principle can be satisfied if the space of <math>n</math> electrons is deemed to be totally ''anti-symmetric'' under the group <math>S_n</math>. Jordan and Wigner (5) introduced creators and annihilators for electrons by operators <math>b_j^*,b_j</math> obeying |
the anti-commutation relations <math>b_j^*b_k+b_kb_j^*=\hbar \delta_{jk}</math>, where now the operators <math>b_j</math> anti-commute | the anti-commutation relations <math>b_j^*b_k+b_kb_j^*=\hbar \delta_{jk}</math>, where now the operators <math>b_j</math> anti-commute | ||
thus: <math>b_jb_k+b_kb_j=o</math>. Again, acting on a vacuum state <math>\Psi_0</math>, this 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. | thus: <math>b_jb_k+b_kb_j=o</math>. Again, acting on a vacuum state <math>\Psi_0</math>, this 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 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 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 Poincare group (the inhomogeneous Lorentz group) was achieved by Wigner and Bargmann 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, and 1940, Pauli, 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 | + | 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 Poincare 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. | ||
| + | |||
| + | |||
| + | 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. | ||
Revision as of 15:54, 15 February 2010
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\[b_jb_k+b_kb_j=o\]. Again, acting on a vacuum state \(\Psi_0\), this 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 Poincare 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.
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.
