Abstract
The observable effects of the extra 1-space and 1-time dimensions of 2T-physics can be found at all scales of physics, large and small, including quantum mechanical domains, in 3+1 dimensions. Often this comes methodically in the form of testable predictions of subtle hidden symmetries and hidden relations among dynamical systems that 1T-physics on its own misses to explain or predict, except for a few cases that were stumbled upon in the past. A couple of surprising examples in non-relativistic physics are discussed in this section, while the important case of conformal symmetry in relativistic field theory , along with some others related by dualities , will be discussed in the following section.
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- 1.
Classical mechanics fails miserably. According to the rules of classical mechanics and classical electricity and magnetism, an accelerating charge must radiate electromagnetic energy, thus losing some of its own momentum and energy. An orbiting electron, as in Fig. 8.1, is accelerating since its velocity changes direction at every instant. Hence it must radiate and lose energy and momentum. Then like an orbiting satellite that is slowing down it should fall to the center of attraction. Estimates, according to classical mechanics, show that the electron would fall into the nucleus, so that the atom would be destroyed in 10−10 s. Evidently this is not true since atoms keep on surviving. This puzzle is resolved by quantum mechanics, which explains that the electron can lose energy by radiating photons if it is in an excited energy state. However, it cannot fall below the ground energy state where it is stable against radiation.
- 2.
There is actually a very small energy difference between the l = 0 and l = 1 states. But it is effectively zero as compared to the other energy differences shown in the figure because it is smaller by a factor of 108. The actual energy difference is known as the Lamb shift, whose experimental measurement was worth a Nobel prize to Willis Lamb. The source of the interaction that splits these energy levels is found in higher order quantum corrections that can be understood and computed correctly in quantum electrodynamics (QED). The accuracy of this, and similar QED computations that are in agreement with experiment to 12 decimal places, is one of the most impressive triumphs of relativistic quantum field theory, showing the degree to which we have managed to understand nature. The fact that the Lamb shift is not zero does not change the conclusions of the hidden symmetry discussed in the text. It only means that in the presence of the quantum corrections the symmetry properties are slightly altered in 1T as well as in 2T-physics.
- 3.
The singleton representation is distinguished by the eigenvalues of the Casimir operators of SO(d, 2). As indicated in Fig. 7.7, the quadratic Casimir eigenvalue is C 2 = \(1-\frac{d^{2}}{4},\) which for SO(4, 2) reduces to C 2 = −3. All other Casimir eigenvalues are also fixed in the singleton representation. It should be noted that the conformal shadow for massless relativistic particles, that has the conformal symmetry SO(4, 2), also has the same C 2 = −3 as can be verified for the case of the massless Klein–Gordon field theory. This is not a coincidence, but is one small manisfetation of the expected duality between the massless particle and the H-atom, predicted by 2T-physics [16].
- 4.
It should be emphasized that unchanging ellipsoidal motion of planets is an approximation for an idealized model of non-relativistic motion in flat space–time. In real life, the perihelion of a planet does change slightly every year relative to other “fixed stars.” There are some very small corrections in the equations that explain this fact, but these can be neglected for the sake of a very simple explanation of the main motion for an isolated Sun–planet system. The ellipse of a planet changes for several reasons. First, the Sun is actually moving within the Milky Way galaxy, so it drags the whole ellipse with it. Second there are other planets nearby whose effects can alter the trajectory of any given planet. Third, and more importantly, as discovered by Einstein, the perihelion of the ellipse for an isolated and idealized Sun–planet system does precess a tiny amount each year due to the modification of the gravitational force explained by general relativity. All of these very small corrections that are incorporated in 1T-physics are also present in 2T-physics. So the SO(4) symmetry discussed in the text, including the fourth dimension, is slightly altered, thus explaining correctly the exact observed motion. However, the conclusions on the fourth dimension arrived at through the simpler approximation remain just as valid after including all corrections.
- 5.
Angular momentum, which is given by the expression \(\vec{L} = \vec{r}\left(t\right) \times\vec{p}\left(t\right),\) is a vector perpendicular to the plane formed by the two vectors \(\vec{r}\left( t\right)\) and \(\vec{p}\left(t\right)\). Here \(\vec{r}\left(t\right)\) is the vector that represents the position of the planet as measured from the Sun. In Fig. 8.4 the vector \(\vec{r}\left(t\right)\) is shown at four different locations, marked by 1, 2, 3, 4 during the course of the year. The vector \(\vec{p}\left(t\right)\) (not shown in the figure) represents the momentum of the planet. At any instant \(\vec{p}\left( t\right)\) is tangent to the ellipse and points in the direction of motion. Both \(\vec{r}\left(t\right)\) and \(\vec{p}\left(t\right)\) lie in the plane of motion. Both vectors \(\vec{r}\left(t\right)\) and \(\vec{p}\left( t\right)\) change as a function of time, as seen in particular for \(\vec {r}\left(t\right)\) at the times when the planet is at locations 1, 2, 3, 4 in the figure. Even though both \(\vec{r}\left(t\right)\) and \(\vec {p}\left(t\right)\) change as a function of time, the angular momentum \(\vec{L}\) computed as the cross product of these two vectors is always independent of time – it does not change in magnitude or direction – its direction in Fig. 8.4 is into the plane at each point on the orbit.
- 6.
The vector \(\vec{A}\) satisfies the equation \(mk\hat{r}\left(t\right) +\vec{A} = \vec{p}\left(t\right) \times\vec{L}\) that is graphically represented in Fig. 8.4 at the four points 1, 2, 3, 4. As seen in the figure, adding the same constant vector \(\vec{A}\) to the changing vector \(mk\hat{r}\left(t\right)\) results precisely in the time dependent vector \(\vec{p}\left(t\right) \times\vec{L}\). Recall that \(\vec{L}\) is another constant vector perpendicular to the plane of the ellipse. Put another way, the vector \(\vec{A},\) given by the expression \(\vec{A} = \vec{p}\left(t\right) \times \vec{L}-mk\hat{r}\left(t\right)\), is constructed as the sum of two time-dependent vectors \(\vec{p}\left( t\right) \times\vec{L}\) and \(-mk\hat{r}\left(t\right) \). Here m is the mass of the planet and k is a constant that is related to Newton’s gravitational constant, while \(\hat{r}\left(t\right)\) is the direction of the vector \(\vec{r}\left(t\right)\). The vector \(\vec{p}\left(t\right) \times\vec{L}\) has a changing magnitude and direction as a function of time, while the vector \(-mk\hat{r}\left(t\right)\) changes direction, but not magnitude as seen at the points 1, 2, 3, 4 in the figure. However, the magic of the hidden symmetry is that their sum remains a constant throughout the motion.
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Bars, I., Terning, J. (2010). Evidence of 4 + 2 as Subtle Effects in 3 + 1 Dimensions. 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_8
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DOI: https://doi.org/10.1007/978-0-387-77638-5_8
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