Fields are divided into two types: "fermionic" and "bosonic". Fermion fields obey the "Pauli Principle", which states that no two fermion field quanta may be in the same quantum state at the same time. Boson fields are not excluded in that way, with the result that "Bose Condensates" of many field quanta in the same state may form. Quantum Field Theory requires that every physically realizable state must be one or the other. This requirement extends to "multiparticle" states composed of identical field quanta.
All fields whose spin quantum number is restricted to odd multiples of one half are fermionic. All fields whose spin quantum number is restricted to integer values are bosonic. In axiomatic Quantum Field Theory, one can prove that a non-trivial field of integer spin cannot anticommute for spacelike intervals (reversal of the order of the product of two operators is the negative of the original product). Similarly, a half-integer spin field cannot commute for spacelike intervals. Since fields which anticommute obey the Pauli Principle, this establishes the relationship between the spin of a field and its fermionic or bosonic nature (its "statistics").
In Quantum Field Theory as applied to the Standard Model of Particle Physics (as currently experimentally verified), the field quanta which make up "matter" fields (such as leptons and quarks) are fermionic, and those which are exchanged in field interactions are bosonic.
One consequence of the exchange of spin one field quanta is that the resulting force can be either attractive or repulsive, depending on the associated charge. This is a property of all theories where odd-integer spin quanta are the medium of the force (ie., electrodynamics). The exchange of even-spin quanta always results in attractive forces. If the attraction is to be proportional to the energy content (as in gravity), the spin must be two.
It is interesting to note that in General Relativity, the particle number current (a flow quantity) associated with fermion fields is manifestly timelike (or null), and their measured energy density may be negative. Additionally, there is a gravitational spin coupling which results in an attractive force for antiparallel spins. The mathematical definition of spin one half fields on an arbitrary manifold constrains the topology of the manifold, and for spins greater than one, there is no well-posed initial value formulation of their dynamics.
Quantum Gravity Concept Map Index:
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©1997, Kenneth R. Koehler. All Rights Reserved.