Abstract
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.
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Notes
- 1.
The clue discussed in [3, 4] is explained in this footnote. The extended superalgebra of M-theory is
$$\left\{ {Q_\alpha, \, Q_\beta } \right\} = P^\mu \left( {\Gamma _\mu } \right)_{\alpha \beta } + Z^{\mu \nu } \left( {\Gamma _{\mu \nu } } \right)_{\alpha \beta } + Z^{\mu \nu \lambda \sigma \rho } \left( {\Gamma _{\mu \nu \lambda \sigma \rho } } \right)_{\alpha \beta }.$$The supercharge Q α is the Dirac spinor of SO(10, 1) with 32 real components, P μ is the momentum in 11 dimensions, while the anti-symmetric tensors Z μν, Z μνλσρ are called the 2-brane and 5-brane charges, respectively. The 528 charges ( P μ, Z μν, Z μνλσρ) commute with each other and with the Q α . This superalgebra is just a rewriting of the following 12-dimensional algebra [3, 4]:
$$\left\{ {Q_\alpha,\, Q_\beta } \right\} = Z^{MN} \left( {\Gamma _{MN} } \right)_{\alpha \beta } + Z^{M_1\, M_2\, M_3\, M_4\, M_5\, M_6} \left( {\Gamma _{M_1\, M_2\, M_3\, M_4\, M_5\, M_6}}\right)_{\alpha \beta}, $$where Q α is the Weyl spinor of SO(10, 2), which also has 32 real components, while Z M N, \(Z^{M_1\, M_2\, M_3\, M_4\, M_5\, M_6}\) are anti-symmetric tensors in 12 dimensions, with a self-duality condition imposed on \(Z^{M_1\, M_2\, M_3\, M_4\, M_5\, M_6}\). It is the reality property of the 32 spinor components that dictates that 2 out of the 12 dimensions must be time-like. The 32 components of the SO(10, 2) spinor Q α and the 528 components of the SO(10, 2) tensors Z MN, \(Z^{M_1\, M_2\, M_3\, M_4\, M_5\, M_6}\) in 12 dimensions exactly reproduce the SO(10, 1) spinor Q α and the SO(10, 1) tensors (P μ, Z μν, Z μνλσρ) when rewritten in an 11-dimensional basis. So the extended superalgebra has a very clear 12-dimensional interpretation. In this way, many key consequences in M-theory that follow from supersymmetry alone can be given a 12-dimensional interpretation with two times. This observation that was made in 1995 begged for an extension of M-theory and is what motivated S-theory in 1996 as a theory in 11-space and 2-time dimensions [5]. S-theory was later instrumental in discovering the fundamental Sp(2, R) gauge symmetry in phase space that gave rise to 2T-physics. 2T-physics, which turned out to stand on its own principles, is applicable to all areas of physics, not only to M-theory. I will return later to the connection between 2T-physics and S-theory or M-theory.
- 2.
C. Vafa, “F-theory”, Nucl. Phys. B469 (1996) 403, [arXiv:hep-th/9602022].
- 3.
C.M. Hull, “Timelike T duality, de Sitter space, large N gauge theories and topological field theory”, JHEP 9807 (1998) 021, [arXiv:hep-th/9806146].
- 4.
The action principle and the global canonical transformations that we wish to extend to local symmetries are best illustrated for a point particle without spin. The action that determines its motion is \(S = \int d\tau \left( {\frac{{dX^M }}{{d\tau }}P_M - H\left( {X, P} \right)} \right)\). A given Hamiltonian H(X, P) specifies a particular system. If we do not focus on a specific system we can ignore H to analyze the general properties of phase space. The first term of the action \(S_0 = \int d\tau \frac{{dX^M }}{{d\tau }}P_M \) dictates the properties of position versus momentum, including their roles in Poisson brackets or quantum commutators. This first term has the symmetries of all canonical transformations. This is a huge global symmetry that includes the special cases mentioned in the text. It is a global rather than a local symmetry on the worldline because its generators do not depend explicitly on the worldline parameter τ which tracks the worldline. Parts of this global symmetry can be promoted to gauge symmetries that are local on the worldline as described in footnotes 5, 6.
- 5.
The laws of motion that include some unique formulas are dictated by the new Sp(2, R) gauge-invariant action principle. The gauge principle requires that every derivative must be replaced by a gauge covariant derivative. In particular, the τ derivative \(\frac{{dX^M \left( \tau \right)}}{{d\tau }}\) that appears in the action in footnote 4 must now be replaced by a covariant derivative. Consequently the first term of the action now gets replaced by \(S = \int d\tau \left\{ {\frac{{dX^M \left( \tau \right)}}{{d\tau }}P_M - \frac{1}{2}A^{ij} \left( \tau \right)Q_{ij} \left( {X, P} \right)} \right\}\) where Q ij(X, P) are the generators of the Sp(2, R)transformations and \(A^{ij} \left( \tau \right)\left( { = A^{ji} \left( \tau \right)} \right)\), a symmetric 2 × 2 matrix, with i, j=1, 2) are the gauge potentials. This modified action now incorporates the consequences of the gauge symmetry and determines the rules of mechanics accordingly. The form of the Q ij (X, P) as a function of (X, P) depends on the forces that are being applied on the particle in d + 2 dimensions. All possible Q ij (X, P) permitted by the Sp(2, R) symmetry are known generally for all physical circumstances. The simplest form occurs when the particle moves in flat space–time X M with no forces other than the symmetry constraints. In this case Q 11=X·X, Q 12=Q 21=X·P, and Q 22=P·P as in Fig. 7.6. In this special case the Sp(2, R) transformations between two perspectives are linear as described in footnote 6. Minimizing the action S with respect to the gauge potential A ij gives the subset of equations of motion which correspond to the constraints Q ij (X, P) = 0 as in Fig. 7.6.
- 6.
In flat space the Sp(2, R) transformations take the special linear form that can be written as a 2 × 2 real matrix of determinant 1, as follows:
$$\left( {\begin{array}{*{20}c} {\tilde X^M \left( \tau \right)} \\ {\tilde P^M \left( \tau \right)} \\\end{array}} \right) = \left( {\begin{array}{*{20}c} {a\left( \tau \right)} & {b\left( \tau \right)} \\ {c\left( \tau \right)} & {\frac{{1 + b\left( \tau \right)c\left( \tau \right)}}{{a\left( \tau \right)}}} \\\end{array}} \right)\left( {\begin{array}{*{20}c} {X^M \left( \tau \right)} \\ {P^M \left( \tau \right)} \\\end{array}} \right).$$The matrix above contains \(a\left( \tau \right), b\left( \tau \right), c\left( \tau \right)\) as its three transformation parameters that are local on the worldline parametrized by τ. Here the doublet \(X^M \left( \tau \right), P^M \left( \tau \right)\) represents the changing phase space along the worldline from the perspective of one observer, while the doublet \(\tilde X^M (\tau),\, \tilde P^M (\tau)\) is the perspective of another observer. The two perspectives are related by the symmetry transformation above, such that \(\tilde X^M \left( \tau \right) = a\left( \tau \right)X^M \left( \tau \right) + b\left( \tau \right)P^M \left( \tau \right)\) and \(\tilde P^M \left( \tau \right) = c\left( \tau \right)X^M \left( \tau \right) + \frac{{1 + b\left( \tau \right)c\left( \tau \right)}}{{a\left( \tau \right)}}P^M \left( \tau \right)\). The gauge fields A ij(τ) must also be transformed as triplets to verify the symmetry of the action in footnote 5. Each observer at any instant τ can choose the values of \(a(\tau),\, b(\tau),\, c(\tau)\) arbitrarily; so the transformation of an observer to a new perspective at instant τ is independent than the transformation of another observer at a different instant \(\tau '\). This freedom of observers is the property of a gauge symmetry that is local on the worldline. When the particle is subject to various forces, the Sp(2, R) transformations from (X M, P M) to \(\left( {\tilde X^M,\, \tilde P^M } \right)\) become more complicated, but they are well understood for all possible cases.
- 7.
To understand this, one must be aware that there is a space–time metric in the definition of the dot products in Fig. 7.6, e.g., \(X \cdot X \equiv X^M X^N \eta _{MN}\), where \(\eta _{MN}\) is the flat space–time metric. So the meaning of X·X is \(X \cdot X = - t^2 + x^2 + y^2 + z^2 - \left( {t'} \right)^2 + \left( {x'} \right)^2 + \cdots\), and similarly \(P \cdot P = - E^2 + p_x^2 + p_y^2 + p_z^2 - \left( {E'} \right)^2 + \left( {p_{x'} } \right)^2 + \cdots\) and \(X \cdot P = - tE + xp_x + yp_y + zp_z - t'E' + x'p_{x'} + \cdots \). Note the minus signs in the two time-like directions t, \(t'\) or E, \(E'\). In the case of 1T-physics the extra terms with \(t', E', x', p_{x'} \) are absent. The equations X·X=P· P=X· P=0 in 2T-physics have solutions only if there are two times; with fewer times these conditions collapse the system to triviality. To see why, consider at first 0-time dimensions, and ask if there are solutions to equations such as \(x^2 + y^2 + z^2 + \left( {x'} \right)^2 = 0\) when there are no minus signs. Then the only solution is trivial x=y=z = x′ = 0, that is vanishing vectors X M=P M=0 which has no physical content. Next consider 1-time, and find that the only solution is that X M∼ P M must be parallel light-like vectors proportional to each other. In that case the angular momentum vanishes L MN=X M P N-X N P M⇒0, and hence this is a trivial solution that does not represent even free motion in 1T-physics. In 2T-physics, the presence of the extra time coordinate permits an infinite number of non-trivial solutions. These solutions represent the Sp(2, R) gauge-invariant sector of phase space. These solutions are parametrized by phase spaces in 1 less time and 1 less space dimensions \(x^\mu,\, p^\mu \), which describe various types of interacting particle dynamics as in the examples of Fig. 7.7. The lower dimensional phase spaces associated with these distinct 1T dynamical systems parametrize the “shadows” of the higher dimensional phase space X M, P M . All the gauge-invariant information, which is guaranteed by the vanishing generators Q ij =0, is captured holographically by each shadow.
- 8.
The more general forms of the constraints Q ij (X, P) = 0, for example, in the presence of background fields for gravity or electromagnetism [23, 24, 43], have been little analyzed. These more general cases would yield their own more interesting shadows.
- 9.
SO(4, 2) is the Lorentz symmetry in 4-space and 2-time dimensions. In the Cartan classification of symmetry groups outlined in footnote 1 of Chapter 5, it corresponds to an analytic continuation of D3.
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Bars, I., Terning, J. (2010). Two-Time Physics. In: Nekoogar, F. (eds) Extra Dimensions in Space and Time. Multiversal Journeys. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77638-5_7
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