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 | + | 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>. |
| − | To get the (nearly) correct energy levels for the hydrogen atom from Heisenberg's new quantum mechanics, Pauli imposed the | + | 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 |
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 | + | 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 |
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. | ||
Dirac introduced his wave-equation for the electron | Dirac introduced his wave-equation for the electron | ||
Revision as of 14:06, 10 February 2010
http://www.mth.kcl.ac.uk/~streater/ The Theorem on Spin and Statistics.
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 \(S_n\). 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 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 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.
Dirac introduced his wave-equation for the electron
