On the Connection between Spin and Statistics
Abstract
In relativistically invariant quantized field theories the following conditions are fulfilled in the normal cases of halfinteger spin connected with exclusion principle (Fermions) and of integer spin connected with symmetrical statistics (Bosons). 1. The vacuum is the state of lowest energy. So long as no interaction between particles is considered the energy difference between this state of lowest energy and the state where a finite number of particles is present is finite. 2. Physical quantities (observables) commute with each other in two spacetime points with a spacelike distance. (Indeed due to the impossibility of signal velocities greater than that of light, measurements at two such points cannot disturb each other.) 3. The metric in the Hilbertspace of the quantum mechanical states is positive definite. This guarantees the positive sign of the values of physical probabilities. There seems to be agreement now about the necessity of all three postulates in physical theories. In earlier investigations I have shown that in the abnormal cases of halfinteger spin connected with symmetrical statistics and of integer spin connected with exclusion principle, which do not occur in nature, not all of the three mentioned postulates can be fulfilled in a relativistically invariant quantized field theory. In this older formulation of the theory for the abnormal cases the postulate (1) was violated for halfinteger spins and the postulate (2) for integer spins, while postulate (3) was always fulfilled. Meanwhile Dirac had directed the attention to the possibility of mathematical theories in which the postulate (3) is abandoned in favoar of more general indefinite metrics in the space of quantum states. In this theory the sum of all probabilities which is conserved in the course of time contains in general also negative terms (``nagative probabilities'') and the square of ``selfadjoint'' operators (which replace the hermitian operators of the usual theory) can also have negative expectation values. Recently Feynman in his ``theory of positrons'', which does not use directly the concept of field quantization but more intuitive methods (which he proves to be equivalent with the former) made the important remark, that the abnormal case of Bosons with spin 1/2 and also of Fermions with spin 0 could be treated in a way similar to the normal case. Considering the effect of an external electromagnetic field (which can produce and annihilate pairs of positive and negative particles) on the initial vacuum, he derived however for the probability that the vacuum is left unchanged a value larger than unity in the abnormal casein contrast to the expected value smaller than one for the normal case. In this paper it is shown for spin 1/2 (SS 2 and 3) and for spin 0 (S 4) that Feynman's treatment of the abnormal cases is equivalent to a different mathematical formulation of the field quantization than which I earlier took into consideration: The new form of the theory for the abnormal cases preserves the postulates (1) and (2) (and also covariance of the theory with respect to chargeconjugation) but violates the postulate (3) introducing ``negative probabilities'' for states with an odd number of negative particles present. Feynman's result of a probability larger than one for the vacuum in such theories is then immediately understandable as the excess above unity of this probability has to compensate the negative probability of states, where one pair is generated. The nonphysical character of these negative probabilities for the abnormal cases is also stressed by the circumstance that the vacuum expectation value of the square of the integral of a component of the current over a finite spacetime region becomes negative in these cases (compare S 3, Eq. (A 48) and (B 49)). In the e^2approximation these vacuum expectation values are indeed very closely connected with the value of the deviation from unity for the probability of the original vacuum in an external electromagnetic field.
 Publication:

Progress of Theoretical Physics
 Pub Date:
 July 1950
 DOI:
 10.1143/ptp/5.4.526
 Bibcode:
 1950PThPh...5..526P