A.7.2

The notion of higher-dimensional space time was encountered in appendix A.4 in the 5-dimensional Kaluza-Klein theory of 1921. The earlier versions of string theory appearing in 1970, namely classical bosonic string theory, employed the Kaluza-Klein idea of compactification to explain the appearance of 26-dimensional space-time naturally arising in this framework. Also quantum bosonic string theory `lives' in 26 space-time dimensions. The introduction of supersymmetry into string theory in 1971 brought the dimensionality of space-time down to 10. 1976 saw the introduction of 11-dimensional supergravity. In 1995 Horava and Witten discovered the eleventh dimension within the context of non-perturbative (strongly coupled) string theory, hinting at U- or M-theory. Even the idea of non-compact dimensions have been re-evaluated within the context of Horava-Witten theory in 1996 and the Randall-Sundrum model (both mentioned at the end of note []) by the possibility of a large fifth dimension giving rise to an effective 5-dimensional action of the universe. In addition, Overduin and Wesson also proposed a non-compact version of the 5-dimensional Kaluza-Klein mechansim in 1997, see appendix A.4.

However, the story doesn't stop with 11 dimensions. Only shortly after M-theory, Vafa introduced 12-dimensional F-theory which addressed some selected problems in M-theory in a natural way (e.g. the vacuum of IIB string theory); [Va96] and [BDS96]. The only problem is that F-theory is (10+2)-dimensional, employing two time dimensions. To avoid the mind-boggling implications of this idea F-theory is considered purely as a mathematical tool with no physical meaning. Continuing along the line of thought introduced in appendix B the non-physicality of mathematical concepts is doubted. This idea also seems to be supported by Vafa: `one can have two views about this 12 dimensional origin [...]: either it is an auxiliary manifold just useful for constructing vacua of string theory [...] or it is more real. In support of the latter interpretation, which we call the ``F-theory'', we point out that this 12-dimensional view-point also solves another puzzle [...]'. For a history of higher dimensions consult [Dur00].

But what exactly is time? It appears to our senses and perceptions of reality that time is a dynamical, ever-flowing `principle'. Time has no location although it permeates all of space. Time is influenced by gravity and velocity, hence destroying the old notion of absolute time. Whereas every spacial dimension has two degrees of freedom, time seems to be locked in a state `motion', flowing from the past to the present, in effect dictating one degree of freedom. Special relativity also only assigns the effects of time to matter: massless particles have, in a relative comparison to the flow of time measured by physical objects, experienced the passage of zero time since the big bang. Although time has such a distinct different status from space, in physics the time and space coordinates are treated on equal footing. Only from `obvious' arguments the special role of time follows. For a very interesting discussion on the difference of space and time (and a possible explanation) consult [Ni99]. But the questions remain:

1. Why does inflation and the general expansion of the universe only affect spacial dimensions?
2. Are there any restrictions on the number of temporal dimensions (e.g. F-theory)?
3. Is time really constrained to only one direction?
4. Couldn't the ideas of compactification be extended to include additional temporal dimension?

Concerning the first question one could argue that such a change in a basic time interval $\Delta t$ would not be physiologically detectable, in the sense that the time dilation in special relativity is only noticeable from a relative (i.e. comparing) point of view. The idea expressed in the second question is interesting if one observes that within the context of string/M-theory the maximal number of spacial dimensions is ten. Additional temporal dimension were probably never really studied before Vafa's proposal. So is it possible to have a (10+n)-dimensional `?'-theory. What about the low-energy limit of these theories: is there (10+2)-dimensional supergravity underlying F-theory? For the third question recall the possibility of negative group velocity mentioned in item three of note []. It is generally so that classical mechanics, electromagnetism and QM do not provide a direction for the passage of time and are invariant under time reversal. $^{\text{\scriptsize\cite{apst1}}}$ Only cosmology $^{\text{\scriptsize\cite{apst1.1}}}$ (with the expansion of the universe) and thermodynamics (i.e. increasing entropy) can define a time arrow (also consult the end of note [16]). Thermodynamics, however, also asks the question of why the universe was highly ordered in the past. The idea of time compactification raised in the fourth question is discussed in [DGS99].

An axiom in QM is that physical observables are generators of linear transformations. Hence within this context energy (the Hamiltonian) generates translations in time. The symmetry statement of classical mechanics concerning the invariance under temporal translations results in the conservation off energy. This ansatz for the possible understanding of time will be taken up below.