String theology

A not inconsiderable portion of September's Physics World, is devoted to string theory. Matthew Chalmers writes a comprehensive survey article on the physics, and Nancy Cartwright/Roman Frigg, from the London School of Economics, write a curiously equivocal article on the philosophy.

Chalmers' article contains the following quote from Leonard Susskind, which caught my attention: "Quantum field theories don’t allow the existence of gravitational forces...String theory not only allows gravity, but gravity is an essential mathematical consequence of the theory."

Now, the claim that quantum field theories don't allow the existence of gravitational forces is slightly misleading. In relativistic quantum theory, each different type of elementary particle corresponds to a Hilbert space equipped with a different unitary, irreducible representation of the Poincare group, (the local symmetry group of space-time). The physically relevant representations of the Poincare group are classified by one continuous parameter, which transpires to be the particle mass, and one discrete parameter, which transpires to be the particle spin. Obtaining these representations is called 'first-quantization'. The graviton, the hypothetical particle of a quantum theory of gravity, corresponds to a particle of mass zero and spin 2. Hence, the graviton corresponds to a well-defined unitary, irreducible representation of the Poincare group in relativistic quantum theory.

From each unitary, irreducible representation of the Poincare group, one can construct another Hilbert space, called a Fock space, and the Fock space is the state space of the quantum field corresponding to the elementary particle in question. This is called 'second-quantization'. Hence, one can construct a well-defined Fock space for the graviton in quantum field theory.

Fock space, however, is the state space for a free field, a field free from interaction with itself or other fields, and there is, as yet, no such thing as a mathematically well-defined quantum field theory of interacting fields. In the case of quantum electrodynamics, one can define interaction Hamiltonians upon Fock space so that, after perturbative calculations have been manipulated by renormalization techniques, reliable scattering amplitudes and cross-sections are obtained. In quantum chromodynamics, even this approach is dubious, and applying such techniques to the quantization of the gravitational field fails completely.

There is, however, another potential approach to the representation of interacting fields. This approach requires Fock space to be abandoned, and for interacting quantum fields to be represented in terms of non-linear mathematical structures. There is a well-defined theory of first-quantized interacting fields, which involves non-linear structures, and just as the second-quantization of a first-quantized free field is a map from a linear vector space into another linear vector space, one would expect the second-quantization of first-quantized interacting fields to be a map from a non-linear space, into another non-linear space. This approach to developing a quantum theory of interacting fields is, I suggest, more likely to bear fruit in the quantization of gravity, than any variation on the string theory programme.