Abstract
We consider scalar field theory defined over a direct product of the real and p-adic numbers. An adjustable dynamical scaling exponent z enters into the microscopic lagrangian, so that the Gaussian theories provide a line of fixed points. We argue that at z = 1/3, a branch of Wilson-Fisher fixed points joins onto the line of Gaussian theories. We compute standard critical exponents at the Wilson-Fisher fixed points in the region where they are perturbatively accessible, including a loop correction to the dynamical critical exponent. We show that the classical propagator contains oscillatory behavior in the real direction, though the amplitude of these oscillations can be made exponentially small without fine-tuning parameters of the theory. Similar oscillatory behavior emerges in Fourier space from two-loop corrections, though again it can be highly suppressed. We also briefly consider compact p-adic extra dimensions, showing in non-linear, classical, scalar field theories that a form of consistent truncation allows us to retain only finitely many Kaluza-Klein modes in an effective theory formulated on the non-compact directions.
Highlights
Quantities like τ and ω as Archimedean, whereas p-adic quantities like x and k will be termed ultrametric.2 We refer to (1.1) as mixed field theory because it is defined over a space which is Archimedean in one direction and ultrametric in another
We argue that at z = 1/3, a branch of Wilson-Fisher fixed points joins onto the line of Gaussian theories
Our results on φ4 theory defined over R × Qp are what one would expect for any continuum field theory in which the dynamics is anisotropic between time and space
Summary
The key point is that when we scale k → pk, the norm and the integration measure scale oppositely:. The natural assignments that make S dimensionless consistent with (1.1)–(1.2) are. We refer to these assignments as engineering dimensions because they describe scalings of the classical action without reference to loop corrections. We see in particular that λ has a positive dimension, meaning that φ4 is a relevant perturbation of the Gaussian fixed point theory, precisely when z > 1/3 — whereas r is always relevant in the same sense, since we require z > 0. For 0 < z < 1/3, our expectation based on power counting is that the Gaussian critical point is the only one available
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