Trying To Conceptualize Two Dimensions Of Time

In summary, Some BSM theories, often but not exclusively string theory inspired, have an extra time dimension, as well as extra spatial dimensions.
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ohwilleke
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Some BSM theories, often string theory inspired, have an extra time dimension, as well as extra spatial dimensions. I'm trying to make sense of what it even means to have more than one time dimension, ideally, with a concrete example.
Some BSM theories, often but not exclusively string theory inspired, have an extra time dimension, as well as extra spatial dimensions.

I'm trying to make sense of what it even means to have more than one time dimension, ideally, with a concrete example that illustrates how the second time dimension would work.

An example of a published paper discussing this concept, which implies that somebody has ultimately figured out the problems identified in the abstract, is:

After the Kaluza–Klein scenario found a home in string theory, physicists became accustomed to the concept of extra space dimensions. Their presence was essential for the mathematical consistency of the theory. Adding extra dimensions of space now seemed natural and even necessary, but physicists shied away from adding also extra dimensions of time.

Why?

Extra time dimensions had actually been quite discouraging to a lot of theorists because extra times bring additional problems that nobody knew how to resolve for many decades.

Itzhak Bars and John Terning, "Two-Time Physics", Multiversal Book Series: Extra Dimensions in Space and Time 67-87 (October 30, 2009) (full text available without a paywall here). A 2001 survey by the first author can also be found here with the following abstract:

Two-time physics (2T) is a general reformulation of one-time physics (1T) that displays previously unnoticed hidden symmetries in 1T dynamical systems and establishes previously unknown duality-type relations among them. This may play a role in displaying the symmetries and constructing the dynamics of little understood systems, such as M-theory. 2T-physics describes various 1T dynamical systems as different d-dimensional `holographic' views of the same 2T system in d + 2 dimensions. The `holography' is due to gauge symmetries that tend to reduce the number of effective dimensions.

Different 1T evolutions (i.e. different Hamiltonians) emerge from the same 2T-theory when gauge fixing is done with different embeddings of d dimensions inside d + 2 dimensions. Thus, in the 2T setting, the distinguished 1T which we call `time' is a gauge-dependent concept. The 2T-action also has a global SO(d,2) symmetry in flat spacetime, or a more general d + 2 symmetry in curved spacetime, under which all dimensions are on an equal footing. This symmetry is observable in many 1T-systems, but it remained unknown until discovered in the 2T formalism. The symmetry takes various nonlinear (hidden) forms in the 1T-systems, and it is realized in the same irreducible unitary representation (the same Casimir eigenvalues) in their quantum Hilbert spaces.

2T-physics has mainly been developed in the context of particles, including spin and supersymmetry, but some advances have also been made with strings and p-branes, and insights for M-theory have already emerged. In the case of particles, there exists a general worldline formulation with background fields, as well as a field theory formulation, both described in terms of fields that depend on d + 2 coordinates.

All 1T particle interactions with Yang-Mills, gravitational and other fields are included in the d + 2 reformulation. In particular, the standard model of particle physics can be regarded as a gauge-fixed form of a 2T-theory in 4 + 2 dimensions. These facts already provide evidence for a new type of higher-dimensional unification.

Some other articles utilizing the concept are (emphasis added):

We discuss the possibility that the cosmological constant problem is solved by raising the number of spatial dimensions a la Kaluza-Klein. In (4+2)-dimensional pure gravity theory with the explicit Λ-term we find classical solutions with vanishing physical cosmological constant and compact 2-dimensions. However, there also exist solutions with the non-vanishing physical Λ-term, and the case Λphys=) is not preferred at the classical level. We conjecture that quantum corrections and/ or additional interactions single out the solution with vanishing physical cosmological constant.

V.A. Rubakov and M.E.Shaposhnikov, "Extra space-time dimensions: Towards a solution to the cosmological constant problem", 125(2-3) Physics Letters B 139-143 (May 26, 1983) and

Phase space and its relativistic extension is a natural space for realizing Sp(2,R) symmetry through canonical transformations. On a (D×2)-dimensional covariant phase space, we formulate noncommutative field theories, where Sp(2,R) plays a role as either a global or a gauge symmetry group. In both cases these field theories have potential applications, including certain aspects of string theories, M theory, as well as quantum field theories. If interpreted as living in lower dimensions, these theories realize Poincaré symmetry linearly in a way consistent with causality and unitarity. In case Sp(2,R) is a gauge symmetry, we show that the spacetime signature is determined dynamically as (D−2,2). The resulting noncommutative Sp(2,R) gauge theory is proposed as a field theoretical formulation of two-time physics: classical field dynamics contains all known results of “two-time physics,” including the reduction of physical spacetime from D to (D−2) dimensions, with the associated “holography” and “duality” properties. In particular, we show that the solution space of classical noncommutative field equations put all massless scalar, gauge, gravitational, and higher-spin fields in (D−2) dimensions on equal footing, reminiscent of string excitations at zero and infinite tension limits.

Itzhak Bars and Soo-Jong Rey, "Noncommutative Sp(2,R) gauge theories: A field theory approach to two-time physics" 64 Phys. Rev. D 046005 (July 20, 2001).

If those papers are too dated, there is a 2008 paper at 77 Phys. Rev. D 125027 by Itzhak Bars entitled "Gravity in Two-Time Physics" with the following abstract:

The field theoretic action for gravitational interactions in d+2 dimensions is constructed in the formalism of two-time (2T) physics. General relativity in d dimensions emerges as a shadow of this theory with one less time and one less space dimensions. The gravitational constant turns out to be a shadow of a dilaton field in d+2 dimensions that appears as a constant to observers stuck in d dimensions. If elementary scalar fields play a role in the fundamental theory (such as Higgs fields in the standard model coupled to gravity), then their shadows in d dimensions must necessarily be conformal scalars. This has the physical consequence that the gravitational constant changes at each phase transition (inflation, grand unification, electroweak, etc.), implying interesting new scenarios in cosmological applications. The fundamental action for pure gravity, which includes the spacetime metric GMN(X), the dilaton Ω(X), and an additional auxiliary scalar field W(X), all in d+2 dimensions with two times, has a mix of gauge symmetries to produce appropriate constraints that remove all ghosts or redundant degrees of freedom. The action produces on-shell classical field equations of motion in d+2 dimensions, with enough constraints for the theory to be in agreement with classical general relativity in d dimensions. Therefore this action describes the correct classical gravitational physics directly in d+2 dimensions. Taken together with previous similar work on the standard model of particles and forces, the present paper shows that 2T physics is a general consistent framework for a physical theory. Furthermore, the 2T-physics approach reveals more physical information for observers stuck in the shadow in d dimensions in the form of hidden symmetries and dualities, that are largely concealed in the usual one-time formulation of physics.

There is an educated layman's description of Bars' research here, which helps a little.

Also, I suspect that my search term looking for papers latched onto Itzhak Bars' research agenda in particular, but I don't think he's the necessarily the only person doing a lot of research on the question. It is a somewhat difficult kind of topic to word search because it is easy to get simplified two dimensional theoretical models mixed in with 3+2 dimensional models.
 
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If we're talking about generalizations of Minkowski space that have more than one timelike dimension, I have some visualization aids, based on the usual pictures of light cones.

First consider the light cone in 1+1 dimensions

https://commons.wikimedia.org/wiki/File:Simple_light_cone_diagram.svg

and in 2+1 dimensions

https://commons.wikimedia.org/wiki/File:Light_cone.png

We use these diagrams to symbolize the light cone in 3+1 dimensions, but taken literally they are depicting 1+1 and 2+1 dimensional space-times.

OK, now consider the "light cone" in 1+2 dimensions:

https://commons.wikimedia.org/wiki/...o_time_dimensions_and_one_space_dimension.pdf

and especially consider that diagram rotated so that the spacelike direction (x coordinate) is horizontal, as in the diagrams for 1+1 and 1+2:



1+2 dimensions is just 2+1 dimensions with spacelike and timelike regions swapped. But it takes a while to appreciate the difference.

2+1 dimensions is a 2d horizontal flatland, with its time evolution proceeding from the bottom of the diagram upwards to the top. Past and future are separated from each other, touching only at one point, the origin of the light cone.

In 1+2 dimensions, space is a 1d horizontal line, and the timelike region surrounds the origin. In particular, past and future are not separated. You can depart the origin into the timelike region, go in a loop and return to the origin, and you have traced out a time loop, a closed timelike curve.

OK, now what about 2+2 dimensions? Consider that you can create the 2+1 light cone by rotating the 1+1 light cone about the vertical axis. In an analogous way, you could create the 2+2 light cone from the 1+2 light cone - but the rotation would be in four dimensions, so it's difficult to visualize.

The main thing is this: In 1+1 dimensions, the spacelike region is divided into two parts, left and right of the origin; but they become connected in 2+1 dimensions, by rotation. Similarly, in 1+2 dimensions, the spacelike direction is divided into a cone to the left and another cone to the right, but in 2+2 dimensions, they become connected again.

So there is a very vague similarity between 2+2 signature, and two linked toruses, if you remember that the toruses stand for a single connected spacelike region and a single connected timelike region.

This should also hold for all spaces of m+n signature, with m and n > 1. We can use the diagram of the 2+1 light cone to visualize any space of the form m+1 dimensions, and the same light cone diagram rotated by 90 degrees to visualize any space of the form 1+n dimensions.
 
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Our intuition that time is different from space stems from the existence of the time arrow. There are various manifestations of the time arrow, but they all originate from the thermodynamic time arrow, i.e. the fact that entropy increases with time. In a hypothetic universe with two or more time-like coordinates, one can define single time arrow as a direction of the entropy gradient (see https://arxiv.org/abs/gr-qc/0403121). Hence even in a theory with multiple times one can have a well defined intuitive notion of a single time.
 
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