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Home , Grand Unified Theory, Quantum geometry, Unified field theory

Journal of Physical Science and Application 7 (4) (2017) 38-46 doi: 10.17265/2159-5348/2017.04.005 D DAVID PUBLISHING

Geometry Is the Common Thread in a Grand Unified Theory

Olaniyi S. Maliki Department of Mathematics, Michael Okpara University of Agricuture, Umudike P.M.B 7267, Abia State, Nigeria

Abstract: The consequence of the wave-particle duality is a pointer to the fact that everything in the , including light and , can be described in terms of particles. These particles have a property called . What the spin of a particle really tells us is what the particle looks like from different directions, in other words it is nothing more than a geometrical property. The motivation for this work stems from the fact that has always played a fundamental role in physics, macroscopic and microscopic, relativistic and non-relativistic. Our belief is that if a GUT () is to be established at all, then geometry must be the common thread connecting all the different aspects of the already known theories. We propose a new way to visualize the concept of four-dimensional space-time in simple geometrical terms. It is observed that our time frame becomes curved, just as the space-frame, in the presence of a massive gravitating body. Specifically, in the event horizon of a black hole, where time seems to grind to a halt for external observers, the time frame appears to curve in on itself, forming an imaginary loop. This results in extreme time dilation, due to the strong gravitational field. Finally we adopt a descriptive view of a GUT called Necklace GUT which attempts to connect gravity together the other three fundamental of , namely the strong, weak and electromagnetic .

Key words: GUT, geometry, space-time, spin, relativity, Einstein field equations.

1. Introduction Geneva, involves a search for novel phenomena predicted by the unified electroweak theory. In The concept of unification began in 1687 with the addition to predicting new physical effects, a unified universal law of gravitation as established by Sir Isaac theory provides a more aesthetically satisfying picture Newton. Since then unification has played a central of how our universe evolves. Numerous physicists role in physics. In the mid-19th century James Clerk share an intuition that, at the deepest level, all physical Maxwell found that electricity and were phenomena match the patterns of some beautiful two facets of . A century later mathematical structure. electromagnetism was unified with the weak nuclear The current best theory of non-gravitational governing radioactivity, in which physicists call forces—the electromagnetic, weak and strong nuclear the electroweak theory. Indeed this quest for force—was largely completed by the 1970s and has unification is driven by practical, philosophical and become familiar as the of particle aesthetic considerations. When successful, merging physics. Mathematically, the theory describes these theories clarifies our understanding of the universe forces and particles as the dynamics of elegant and leads us to discover things we might otherwise geometric objects called Lie groups and fiber bundles. never have suspected. Much of the activity in It is, however, somewhat of a patchwork, this is experimental today, at accelerators because a separate geometric object governs each such as the Large at CERN near force. Over the years physicists have proposed various Corresponding author: Olaniyi S. Maliki, associate Grand Unified Theories, or GUTs, in which a single professor, research fields: , mathematical geometric object would explain all these forces, but no modeling, computational mathematics, .

Geometry Is the Common Thread in a Grand 39 one yet knows which, if any, of these theories is true. particle really tells us is what the particle looks like One would expect that in a fully unified theory, from different directions. For example, a particle of gravity and matter should also combine naturally with spin 0 is like a dot, it appears the same from every the other forces, all as parts of one mathematical direction. On the other hand, a particle of spin 1 is like structure. Since the 1980s , the dominant an arrow, it appears different from different directions. research program in theoretical particle physics has It is only when it is rotated a complete revolution (360 been an attempt to describe gravity and the Standard degrees) does the particle appear the same. A particle Model using elaborate constructs of strings and of spin 2 can be visualized as a double-headed arrow, membranes vibrating in many dimensions. it appears the same if it is rotated half a revolution String theory is however not the only effort. Loop (180 degrees). Similarly, higher spin particles appear as an alternative, uses a more the same if rotated through smaller fractions of a minimal framework, and is closer to that of the complete revolution. Remarkably, it turns out that Standard Model [1]. Building on its insights, the E8 there are particles that do not appear the same if theory [2] was proposed as a new unified theory in rotated through just one revolution: they must be 2007. Here the basic idea is to extend Grand Unified rotated through two complete revolutions! Such Theories and include gravity as part of a consistent particles are said to have spin ½. It is clear here that geometric framework. In this unified field theory, all the concept of spin is nothing but a geometrical forces and matter are described as the twisting of a property of the particle under consideration. single geometric object [3]. 4. Importance of Relativity Theory 2. From Electromagnetism to Geometry A remarkable consequence of relativity [4] is the The geometric view of nature follows naturally way it has revolutionized our ideas of space and time. from the way the world around us works. The simplest We now know that there is no such thing as absolute and most familiar examples are the forces of time. On the other hand, one major prediction of electricity and magnetism. Electric sparks, magnetic [5] is that time should appear to be attraction and laser light are different manifestations slower near a massive body like the earth. This is of the electric and magnetic fields that pervade space. because there is a relation between the of light Physicists do believe that everything in the world—all and its frequency, the greater the energy, the higher the forces of nature and even all the particles of the frequency. As light travels upward in the earth’s matter—arises from different kinds of fields. The gravitational field, it loses energy, and so its frequency behaviour of these fields suggests an underlying goes down. This implies that the length of time geometric structure. between one wave crest and the next goes up. To an observer high up, it would appear that everything 3. Elementary Particles and Geometry down below was taking longer to happen. Although Using the wave-particle duality, it is now evident light is made up of waves, Planck’s quantum that everything in the universe, including light and hypothesis tells us that in some ways it behaves as if it gravity, can be described in terms of particles. These were composed of particles: it can be emitted or particles have a property called spin. One way of absorbed only in packets, or quanta. Equally, thinking of spin is to imagine the particles as little Heisenberg’s implies that tops spinning about an axis. However, they do not particles behave in some respects like waves: they do really have any well-defined axis. What the spin of a not have a definite but are “smeared out” with

40 Geometry Is the Common Thread in a Grand Unified Field Theory a certain probability distribution. There is thus a   xx duality between waves and particles in quantum  The Jacobian of the transformation J  , and its theory. For some purposes it is helpful to think of inverse  , are given by: J particles as waves and for other purposes it is better to xx think of waves as particles. JJJJ ,  ,   xx 5. Notation 6. Remark will denote the 4-dimensional spacetime , g  The components of a tensor may not be of GR, R is the real line and g is the . independent. If for example A A then we say An event in ,g is represented by the that A is a symmetric tensor, and if B B , components of the contravariant position vector: then A is antisymmetric.  0123 x  xxxx,,,  Einstein summation convention applies throughout this work. In the Cartesian basis the components of this vector 2    The quantity ds g dx dx is referred to as are the time coordinate, x0  t , and the spatial  g a line element and  is the metric tensor. coordinates, xi xxx123,, xyz ,, . 7. The Flow of Spacetime Differentiation of a function F with respect to the An elegant way of visualizing the curvature of  F space-time coordinate x is written as F . 0123  x spacetime  x ,,,xxx , is illustrated in Fig. 1. Here Performing a coordinate transformation simply spacetime is composed of time axis pointing upwards computes the values of the coordinates at that event, in time and carrying with it at each point a triad 123 from a different reference frame; this different frame  x ,,xx subsequently referred to as the spaceframe. could correspond to an observer that is accelerating In the vicinity of a massive object both the time frame relative to the first observer, or where the second and the space frame become curved as shown. Hans observer is at a different point in a gravitational field. von Baeyer [6], in a prize-winning essay described

The coordinate transformation is: spacetime as an invisible stream flowing ever onward,

Time axis x0  t B x00112233 dx,, x dx x dx , x dx 

x3

A xxxx0123,,,  O’ x2 x1 x3 World line of a particle in a O x2 gravitational field x1

Fig. 1 The flow of curved spacetime.

Geometry Is the Common Thread in a Grand Unified Field Theory 41

bending in response to objects in its path, carrying 2 ij dsij dx dx (3) everything in the universe along its twists and turns. An important feature of the Kronecker-delta is that The elemental distance between the two its determinant is positive definite. A manifold infinitesimally separated points A and B is given by endowed with a metric whose determinant is the line element: positive-definite is called a Riemannian manifold. It is 2   important to know that the metric actually defines the ds g dx dx (1) geometry of a space. The curvature of the region of space is specified by Our definition here coincides with the Minkowski’s the metric tensor g . four-dimensional spacetime if we take; The case of flat spacetime Fig 2, is similar except 00 xitdxidti  , 1 for the fact that there is no curvature, and this situation 022 corresponds very well with the standard Minkowski’s Hence dx   dt and 22222 four-dimensional space. In flat spacetime a particle ds dx dx dt dx dy dz moves from point A to a point B with coordinates as shown. The observer in his frame of reference initially with  diag 1,1,1,1 . at O, monitors the particle until it reaches point B by which time his frame has traveled upwards to reach 8. Properties of the Einstein Field Equations 0 O in time by dx dt . We define the line We write the Einstein Field equations of general element: relativity as: 2022 1 2 22 3 GT  (4) ds dx dx dx dx (2)  where GR  1 g R is the Einstein tensor, as the elemental distance (squared) between the two  2   is the Ricci tensor. T is the infinitesimally separated points A and B. By RgR   introducing the Kronecker-delta energy-momentum tensor   8Gc4 , G being the

ij 0, ij, 1 ij , the above can be Newtonian gravitational constant. At this point we written more compactly as: consider the following three basic properties of Eq. (4):

Fig. 2 The flow of flat spacetime.

42 Geometry Is the Common Thread in a Grand Unified Field Theory

(a) Eq. (4) is a tensor equation. This is necessary so, quantities. since the principle of invariance under coordinate 9. Exact Analytic Solutions of Einstein’s transformations must hold, in other words the Field Equations equations of physics must look the same in any frame of reference. No general solution to Einstein’s field equations, (b) We can interpret Eq. (4) more simply as: Eq. (4), is known to exist. However, solutions are Tensor representing geometry of space = Tensor known for rather specific configurations. The simplest representing energy content of space solution is for a spacetime which is completely empty i.e., it is the presence of matter (or energy) in space (of gravitating matter) is called a vacuum, and whose that distorts the neighbouring geometry. Most metric is given by the Minkowski metric, viz: equations of mathematical physics can be interpreted 22222 similarly. ds dt  dx  dy  dz (c) The solution to Eq. (4) is a geometrical object, The next simplest metric is that for a spacetime namely a line element given by: containing a homogeneous and isotropic fluid, whose 2   ds g dx dx energy-momentum tensor is of the form: where g  is the metric tensor to be solved for in Eq. TuuP    (5) (4). GR (General Relativity) is one of the greatest where    tPPt,    , the time dependent advances in from the past century. density and pressure respectively, and is given by the Remarkably, was able to understand Freidmann-Robertson Walker’s solution, that the geometry of the universe is determined by, 222222 ds dt  a t dx  dy  dz  (6) and responds to, the gravitating content of the universe. Einstein proceeded to construct the The metric of a spacetime [8] which in the limit dynamical rules for spacetime with arbitrary geometry; tends to a vacuum solution at infinity and containing a these rules are precisely the Einstein’s field equations, single stationary black hole of mass M at the origin, is Eq. (4) of GR. It became evident that gravity is just a given by Schwarzschild’s solution [9], manifestation of the location-dependence of the metric 1 2222222MM ds 11 dt  dr  r d (7) g  . The field equations determine the values of the rr components of the metric of a spacetime for some dd2222 sin d is the solid angle element, and known content. Once the metric is known one can M is the mass of the blackhole. begin to compute geodesics in the spacetime; these are The metric for a spacetime containing a stationary the paths that bundles of light rays travel along or the black hole of mass M in a universe containing a orbits that planets trace out. This enables GR, as a  is given by Kottler’s gravitational theory, to predict directly solution [10]:

1 222222222MM ds11   r dt    r dr  r d  (8) rr33

This is the well known Schwarzschild de-Sitter introduced by Einstein deliberately to cancel out the metric. The cosmological constant  [7], was expansion of the universe, predicted by the field

Geometry Is the Common Thread in a Grand Unified Field Theory 43 equations, at the time this did not make sense. is also indirectly as a result of the fact that light cannot However, Edwin Hubble [11] in 1929, proved that the escape from the black hole. In the case of Kottler’s universe was indeed expanding. solution incorporating a cosmological constant  , A single black hole of mass M immersed in a the event horizon will be given by the real roots of the homogeneous isotropic fluid has a metric given by cubic: McVittie’s solution [12, 13]: rrM3 36  0 (10) In this case there is a possibility of three 2atr  M ds22 dt  singularities. For McVittie’s solution, time disappears 2atr  M  for rMat 2   , however this does not give rise to 4 a singularity in the space frame, though it is curved to 2222M at1 drrd (9) a certain degree, and this can be measured. 2atr 10. Quantum Gravity and Unified Theories The important thing to note in each of the above solutions is the fact that the coefficient of the time The unification of general relativity and quantum elemental squared is non constant and this leads us to into a theory of quantum gravity is of great conclude that the time frame is not flat, but naturally interest to theoretical physicists and mathematicians. curved just as the space frame is. This project is vigorously being pursued in numerous In the case of the Schwarzschild’s solution we have scientific institutes and laboratories all around the a singularity at rM 2 in the space frame, but in the world. If successful, it should also provide a time frame, time disappears for this same value. Now consistent framework for incorporating all other rM 2 is the event horizon in Fig. 3: a region in the fundamental forces of nature. Unfortunately despite vicinity of the stationary black hole where time seems intense efforts over the last years it is far from clear at to grind to a halt. this time what a consistent theory of quantum gravity Our proposition here is that the time frame has will look like, and what its main features ought to be. completely curved in on itself, forming a kind of In the light of these uncertainties, the best strategy imaginary loop, resulting in extreme time dilation, due appears to be one which is both diversified and to the strong gravitational field of the black hole. This interdisciplinary. To that extent, it would make sense

Fig. 3 The event horizon.

44 Geometry Is the Common Thread in a Grand Unified Field Theory

to incorporate all the major current approaches to quantum gravity, in particular [14] and string theory [15] and their modern developments, as well as canonical quantization (e.g. [16]) and discrete models of quantum gravity. We do not intend to go into the theory of quantum gravity here, however it is important to note that the current canonical approaches to this theory, emphasize the geometrical aspects which appear well suited to deal with unsolved conceptual issues of quantum Fig. 4 The quantum plane ()xyyx . gravity, such as for example, the problem of time or the interpretation of the of the universe In the quantum plane we replace the property [17]. xyyx by xyqyx where q is some parameter. We no longer have points, however we can continue 11. Quantum Geometry to work algebraically with x and y. Every geometry is associated with some kind of Define x, yxyyx   , the commutator bracket. space. Quantum (or noncommutative) geometry Hence xyqyx can be rewritten as: [18-20] deals with quantum spaces, including the x,1yqyx classical concept of space as a very special case. In    classical geometry spaces are always regarded as The commutative case is obtained when q 1. collections of points equipped with the appropriate 13. Summary and Conclusion additional structure (as for example a topological structure given by the collection of open sets, or a From the foregoing, it is clear that our objective was smooth structure given by the atlas). In contrast to not to postulate a GUT, but rather to examine existing classical geometry, quantum spaces are not efforts towards this goal and the inherent difficulties. interpretable in this way. In general, quantum spaces More importantly we emphasize that geometry is have no points at all! They exhibit non-trivial intrinsic to all aspects of physics. Our view is that a quantum fluctuations’ of geometry at all scales. successful construction of a GUT must incorporate Quantum geometry as constructed by Alain Connes geometry as the common thread connecting all the [21] is a well developed field of active research with different aspects of the already established theories. interesting applications. What follows is a simple It is now a general belief that all four of the example of what quantum geometry is. fundamental forces of nature are, in fact, the manifestations of a single unifying force which has 12. The Quantum Plane yet to be discovered. Just as electricity, magnetism, A simple example of quantum geometry is the and the weak force were unified into the electroweak quantum plane (Fig. 4). Usually, a plane is described , they work to unify all of the fundamental by two coordinate functionsx, y . Naturally, the forces. From our current understanding of physics, functions xyyx and are the same since it does not forces are not transmitted directly between objects, but matter whether you measure x first and then y or y first instead are described by intermediary entities called and then x. This is precisely what is lost in the fields. All four of the known fundamental forces are quantum world. mediated by fields, which in the Standard Model [18]

Geometry Is the Common Thread in a Grand Unified Field Theory 45

Geometry

Strong interaction

Gravitational Electromagnetic interaction interaction

Fig. 5 Quantum Necklace GUT. of particle physics result from exchange of gauge mathematically rigorous way that quantum geometry bosons. Specifically the four interactions to be unified does indeed unify all the fundamental forces of nature. are: References  : the interaction responsible for holding together to form , and [1] Mann, R. 2010. An Introduction to Particle Physics and the Standard Model. Boca Raton: CRC Press. ISBN: holding and also together to form 978-1-4200-8298-2. nuclei. The exchange particle that mediates this force [2] Adams, J. 1996. Frank Lectures on exceptional Lie is the . Groups, Chicago Lectures in Mathematics. University of  Electromagnetic interaction: the familiar Chicago Press, ISBN 978-0-226-00526-3, MR 1428422. [3] Frankel, T. 1997. The Geometry of Physics: An interaction that acts on electrically charged particles. Introduction Cambridge University Press. The is the exchange particle for this force. [4] Einstein, A. 1952. The Meaning of Relativity. London and  Weak interaction: a short-range interaction New York: Routledg. [5] Dirac, P. A. M. 1996. General Theory of Relativity. responsible for some forms of radioactivity, which Princeton: Princeton University Press. acts on , , and quarks. It is governed [6] Baeyer, H. V. 2001. On Close Inspection (Spacetime by the . Foam). The Sciences.  Gravitational interaction: a long-range [7] Lematre, G. 1931. “Expansion of the Universe: A Homogeneous Universe of Constant Mass and Increasing attractive interaction that acts on all particles. The Radius Accounting for the Radial Velocity of postulated exchange particle has been named the Extra-Galactic Nebulae.” Monthly Notices of the Royal . Astronomical Society 91 (483): 490. [8] Kiefer, C. 2007. Quantum Gravity. Oxford University We also recall that the quest to unify gravity with Press. ISBN 0-19-921252-X. the other three fundamental forces is known as [9] Schwarzschild, K. 1916. “On the Gravitational Field of a quantum gravity. While research efforts are still Mass Point according to Einstein’s Theory.” Sitzungsber. ongoing in this field, we adopt the following Preuss. Akad. Wiss. Berlin (Math. Phys.). 189: 196. [physics/9905030]. descriptive view of a GUT which we call Quantum [10] Kottler, F. 1918. “ber die physikalischen grundlagen der Necklace GUT, as shown in Fig. 5. einsteinschen gravitationstheorie.” Annalen der Physik In the above figure we propose Quantum Geometry 361 (14): 401-62. as described in Section 11, as a good candidate for the [11] Hubble, E. P. 1937. The Observational Approach to Cosmology. Clarendon Press. LCCN 38011865. geometry connecting all the fundamental forces of [12] McVittie, G. C. 1966. “An Example of Gravitational nature. Our research focus for the future is to show in a Collapse in General Relativity.” Astrophysical Journal 143

46 Geometry Is the Common Thread in a Grand Unified Field Theory

(3): 682. Lecture Notes in Physics 631: 41-135. [13] Kaloper, N., Kleban, M., and Martin, D. 2010. [19] Maliki, S. O., and Ugwu, E. I. 2014. “On q-Deformed “McVittie’s Legacy: Black Holes in an Expanding Calculus in Quantum Geometry.” Applied Mathematics 5: Universe.” Phys. Rev. 81: 104044. 1586-93. [14] Baeyer, H. V. 2001. On Close Inspection (Spacetime [20] Maliki, S. O. 2007. “A Gentle Introduction to ). The Sciences. Geometry.” A postdoctoral seminar presentation at the [15] Joseph, P. 1998. String Theory. Cambridge University African Institute for Mathematical Sciences (AIMS) Press, ISBN 0521672295. Muizenberg, South Africa (AIMS_Annual [16] D 13: 3214-8. _Report_2007-8.pdf). Institute for Mathematical Sciences [17] Hawking, S. W., and Hartle, J. B. 1983. “The Wave (AIMS) Muizenberg, South Africa (AIMS_Annual Function of the Universe.” Physical Review D 28: _Report_2007-8.pdf). 2960-75 [21] Connes, A. 1994. . New York: [18] Thiemann, T. 2003. “Lectures on Loop Quantum Gravity.” Academic Press.