Physics With Two Time Dimensions Jacob G. Foster 1 and Berndt M¨ uller 2 $^{1}$Complexity Science Group, Department of Physics, University of Calgary, Calgary, Alberta, Canada T2N 1N4 $^{2}$Department of Physics & Center for Theoretical and Mathematical Science, Duke University, Durham, NC 27708, USA (Dated: January 25, 2010) We explore the properties of physical theories in space-times with two time dimensions. We show that the common arguments used to rule such theories out do not apply if the dynamics associated with the additional time dimension is thermal or chaotic and does not permit long-lived time-like excitations. We discuss several possible realizations of such theories, including holographic representations and the possibility that quantum dynamics emerges as a consequence of a second time dimension. Modern unified theories of the fundamental interac- tions generally posit that we live, at microscopic scales, in a universe with more than three spatial dimensions. Following Kaluza’s [1] and Klein’s [2] original idea, the extraneous dimensions are assumed to be compactified so that excited modes of the elementary particle fields involving nontrivial dynamics in these dimensions are ex- ceedingly massive and therefore not directly observable. The sole exception to this rule may be gravity [3]. field theory, constructed as the boundary field theory of a six-dimensional Anti-deSitter space-time with two time dimensions containing a Schwarzschild black-hole. In Sect. IV we point out that, once the notion of field theories in space-times with two time dimensions is ac- cepted as physically viable, the quantum dynamics of the reduced field theory in Minkowski space can be gener- ated by the mechanism of micro-canonical quantization from the underlying classical dynamics of a field in a higher-dimensional space-time that contains two time di- mensions. Finally, we summarize our results and present some thoughts about the possible cosmological context of a second time dimension. The Kaluza-Klein concept is naturally realized within superstring theory, which requires at least six additional dimensions for internal consistency [4]. It has been noted in recent years that the additional spatial dimensions do not have to be aspects of a genuine space-time contin- uum, but could be artefacts of the algebraic structure of some underlying quantum mechanical system [5]. One common feature of most theoretical speculations of this kind is that the additional dimensions are as- sumed to be spatial. The reason for this restriction is that physics in the presence of more than one temporal dimension is generally considered to violate several fun- damental and well established properties of nature. The two most important of these principles are causality and unitarity, but there are also concerns that additional time dimensions imply tachyonic modes and ghost fields, i. e. field components with negative norm. Let us briefly review why the basic principles of causal- ity, unitarity, and vacuum stability are thought to be vi- olated in the presence of additional time dimensions. We will discuss the case against more than a single time dimension in Sect. II, followed by an overview of the arguments that have led us to reconsider this case. In Sect. III, we first review the mathematical difficulties as- sociated with the initial-value problem for wave equations in spaces with multiple time dimensions. After explain- ing why a recently proposed resolution of the ill-posed na- ture of the initial-value problem does not work in general for interacting fields, we will argue that interacting fields without propagating time-like modes (with respect to the additional time dimension) can evade this difficulty. In particular, we show how the difficulty can be circumnavi- gated if the field is thermalized in the additional time di- mension. We propose a specific model of such a quantum 1. Causality: In space-times with at least two time di- mensions, it is always possible to construct closed time-like curves [6]. This is especially worrisome in the context of classical physics, where observers are not part of the physical system and can “act” in un- predictable ways. The existence of closed time-like curves implies that an observer can revisit the past and, if we accept the tenet of “free will”, change it in a manner that is incompatible with the already experienced future. The classic example of a result- ing inconsistency is the trip back in time, where the hero unwittingly kills his own ancestor (the grand- father paradox). Obviously, anything resembling the common notion of causality cannot be main- tained under such circumstances. The question of the well-posed nature of the Cauchy (initial-value) problem in space-times with closed time-like paths was investigated in detail by Aref’eva et al. [9]. 2. Unitarity: While violations of causality can arise both in classical and quantum dynamics, violations arXiv:1001.2485v2 [hep-th] 25 Jan 2010 II. MULTIPLE TIME DIMENSIONS A. The case against multiple time dimensions I. INTRODUCTION