Quantum field theory in non-integer dimensions
In a 1973 paper entitled Quantum Field-Theory Models in Less Than 4 Dimensions, Kenneth G. Wilson studied field-theories for spacetime dimension d between 2 and 4. With unconventional renormalizations, these models were found to have non-Gaussian ultraviolet renormalization group fixed points. Wilson's method was perturbative dimensional regularization: the Feynman-graph integrals were analytically continued to non-integer d. His work left open the question of the nonperturbative existence of the models. Since that landmark paper, Yuval Gefen, Amnon Aharony and Benoit B. Mandelbrot have shown that Ising spin models on fractal lattices have critical properties like those predicted for non-integer dimensions by the analytic continuation, or {epsilon}-expansion method. This work shows that fractal lattices and continua provide also a nonperturbative definition of field-theories in non-integer dimensions. The fractal point-sets employed are the Sierpinski carpets and their higher-dimensional generalizations. This class of point-sets has a tunable dimension which allows the approach to four from below. Furthermore, the carpets have discrete groups of scale or dilation invariances and infinite order of ramification. A class of scalar field models are defined on these sets which should reduce to the standard models when d {nearrow}4. The propagator for these models is given by a proper-time or heat-kernel representation. For this propagator, reflection-positivity is established, a general scaling law is conjectured (and established in a special case), and the perturbative renormalizability shown to be governed by the spectral dimensionality. Scalar models with another choice of propagator, the hierarchical propagator, are studied by rigorous renormalization-group methods.
- Research Organization:
- Ohio State Univ., Columbus, OH (USA)
- OSTI ID:
- 5467965
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
Similar Records
Conformal sector of quantum Einstein gravity in the local potential approximation: Non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance
Quadratic divergences in dimensional renormalization of 1/N expansions
Related Subjects
QUANTUM FIELD THEORY
PERTURBATION THEORY
ANALYTICAL SOLUTION
DIMENSIONS
FRACTALS
ISING MODEL
PROPAGATOR
QUANTUM GRAVITY
RENORMALIZATION
SCALAR FIELDS
SCALING LAWS
SPACE-TIME
CRYSTAL MODELS
FIELD THEORIES
MATHEMATICAL MODELS
645400* - High Energy Physics- Field Theory