Preons from holography

Transcription

1 arxiv: v1 [physics.gen-ph] 23 Jan 2008 Preons from holography T. R. Mongan 84 Marin Avenue Sausalito, CA USA (415) tmongan@gmail.com January, 2008 Abstract Our universe is apparently dominated by a vacuum energy/ cosmological constant. The holographic principle then indicates only a finite number of bits of information will ever be available to describe the universe. In that case, the Standard Model (SM) of particle physics based on continuum mathematics with infinite degrees of freedom can only approximate the underlying finite dimensional reality. That raises the question of the relation between SM particles and the bits of information in a holographic description of the universe. This paper outlines a wrapped preon model based on a holographic representation of a closed vacuum-dominated universe. The wrapped preon model suggests the fundamental elements of particle physics are strands rather than point particles, because points can t be wrapped. The model produces SM particles, and only those particles. As an approximation, SM particles in the wrapped preon model can be identified with preon bound states in non-local dynamics based on three-preon Bethe- Salpeter equations with instantaneous three-preon interactions. Introduction Our universe is apparently dominated by a vacuum energy/cosmological constant. The holographic principle then indicates only a finite number of bits 1

2 of information will ever be available to describe the universe. In addition, quantum mechanics is known to be non-local. The Standard Model (SM) adequately describes particle physics within the universe. It involves three generations of fermions, with forces between fermions generated by boson exchange. However, the SM is a local continuum theory involving infinite degrees of freedom, so it can only approximate an underlying non-local finite dimensional reality. That raises the question of the relation between SM particles and the bits of information in a holographic description of the universe. This paper outlines a wrapped preon model, based on a holographic description of the universe, producing SM particles and only those particles. It does not address the more difficult problem of developing a holographic finite dimensional lattice gas model of interactions between SM particles underlying the existing continuum SM. The wrapped preon model was motivated by two ideas from Bilson- Thompson s (BT) preon model [1]. The first is the concept that preons are strands and not point particles. The second is the hint from the BT fermion states that spin relates to wrapping of preon strands around each other. The model assumes SM particles are composed of three preon strands, each characterized by two bits of information, bound by non-local three body interactions. The two bits of information on a strand are identified with two of the bits of holographic information on an observer s horizon, constituting all information available to describe physics within the four-dimensional spacetime of a vacuum-dominated closed universe. The wrapped preon model is consistent with recent efforts to base particle physics on one-dimensional objects, to avoid the rampant infinities in theories based on point particles. Background on holography Before outlining the wrapped preon model, some background about the holographic principle is necessary. The holographic principle says all information that will ever be available to any observer about physics within a horizon is given by the finite amount of information on the horizon, specified by one quarter of the horizon area in Planck units. Consistent with the model in Ref. 2, the universe appears to be dominated by vacuum energy density ε ν related to a cosmological constant Λ cm 2 by ε ν = Λc4. Therefore, 8πG the universe is asymptotic to a de Sitter space, with an event horizon at distance R H = 3/Λ = cm. The holographic principle then says 2

3 only N = 3π = 5 Λδ 2 ln bits of information ( 3π degrees of freedom) will Λδ 2 ever be available to describe the universe. So, existing theories involving continuum mathematics can only approximate an underlying finite dimensional holographic theory involving bits of information. Quantum mechanics is known to be non-local, but the mechanism of nonlocality has remained obscure. A holographic quantum mechanical theory is essentially non-local if the wavefunction on the horizon (describing the evolution of all information available about the universe) is the boundary condition on the wavefunction describing the information distribution within the horizon. Then any quantum transition altering the wavefunction on the horizon is reflected in an instantaneous change in the wavefunction describing the information distribution within the universe. A quantum mechanical description of the bits of information on the de Sitter horizon requires wavefunctions specifying the probability distribution of those bits of information. A quantum description of the total amount of information eventually available about the universe can be obtained [3] by identifying each area (pixel) of size 4δ 2 ln 2 on the de Sitter horizon with one bit of information on the horizon at radius R H = 3/Λ. Consider a wavefunction specified in a spherical coordinate system centered on the observer s position. The z-axis of such a coordinate system pierces the i th pixel of area 4δ 2 ln2 on one hemisphere of the horizon and the j th pixel of area 4δ 2 ln2 on the opposite hemisphere of the horizon. Suppose θ i is the polar angle measuring the angular distance from the z-axis through the i th pixel to the point on the horizon where the wavefunction is evaluated. Then wavefunctions of the form cosθ i on the horizon have wavelength 2πR H and can define the probability distribution for finding the information associated with the diametrically opposed (i, j) pair of bits at any location on the horizon, with the maximum probability of finding information associated with the diametrically opposed (i, j) bits in the two pixels where the z axis pierces the 3 horizon. If the z-axis of a wavefunction 2πR H cosθ i is associated with each pair of diametrically-opposed bits of information on opposite hemispheres of the horizon, state identifiers (0, 0), (0, 1), (1, 0) and (1, 1) specify the two bits of information associated with the wavefunction for any quantum state on the horizon. In an observer s coordinate system used to describe all of the information on the horizon, a quantum state with zero in the pixel on the horizon in one radial direction and one in the pixel on the diametrically opposite hemisphere is not identical to a state with one in the first pixel 3

4 and zero in the pixel on the opposite hemisphere. So, quantum states with identifiers (1, 0) and (0, 1) are not identical on the horizon. Strands in the wrapped preon model Identify each of the N/2 diametrically-opposed pairs of bits on the horizon with the wavefunction for a single preon strand within the horizon. The wavefunction for the two bits of information on the horizon is the boundary condition on the wavefunction specifying the probability distribution for finding the associated strand somewhere within the horizon. Charge is introduced by identifying 1 with charge e/6 and 0 with charge e/6. Charge neutrality of the universe requires equal numbers of e/6 and e/6 charges. Also, spontaneous production of the universe by quantum fluctuation from nothing [2] requires total quantum numbers equal zero, charge neutrality in the universe, and equal numbers of e/6 and e/6 charges. In the wrapped preon model, wavefunctions for quantum states on the horizon with state identifiers (0, 0), (0, 1), (1, 0) and (1, 1) are boundary conditions on wavefunctions specifying the probability of finding strands [with charges ( e/6, e/6), ( e/6, e/6), (e/6, e/6) and (e/6, e/6), respectively, on their ends] within the horizon. Two bits on diametrically-opposed pixels on the horizon specify a strand that can be bent through an angle modulo 2π radians. So, a pair of bits can represent a straight strand, a curved strand, a spiral strand with one full turn, or (in the case of a neutral strand with opposite charges on each end) a closed loop. Open charged strands bound into massive SM particles have lengths of the order of a characteristic hadronic electric charge radius. Neutral strands bound into massless vector bosons have lengths of the order of the boson wavelength. Strand interactions The wrapped preon model assumes non-local three body forces bind three, and only three, strands into SM particles. The model also involves three types of strand interactions within SM particles. First, pairs of ( e/6, e/6) and (e/6, e/6) strands, with a net neutral charge, do not exist within a single SM particle because they immediately convert to pairs of ( e/6, e/6) and (e/6, e/6) strands. Second, there are one strand interactions where oppo- 4

5 sitely charged ends of neutral strands join to form neutral loops and the reverse process where neutral loops break open to recreate neutral strands. The states (0, 1) and (1, 0) represent open neutral strands with charges ( e/6, e/6) or (e/6, e/6) on their ends or closed neutral loops formed by joining oppositely charged ends of open neutral strands. Third, there are two strand interactions where oppositely charged ends of two open strands join to create a single open strand and the reverse process where a single composite open strand breaks in the middle to recreate two open strands. The second and third interactions are necessary for creation and dissociation of bosons responsible for forces between SM particles. The above interactions are both necessary and sufficient to produce SM particles (including three generations of fermions), and only those particles. No other interactions are needed to explain SM particles and their interactions. So, to avoid unnecessary multiplication of entities, additional interactions are not included in the model. Standard Model particles In the wrapped preon model, all SM particles are composed of three preon strands. Bosons exchanged in the crossed channel of fermion-fermion scattering are three preon structures that must conserve charge and angular momentum, so bosons must have zero or unit charge. In this model, spin results when one strand wraps around another and strand configurations corresponding to SM particles are sketched in Figure 1. For simplicity, Figure 1 represents unwrapped open strands as straight line segments and spin zero bosons are shown as three parallel unwrapped open strands. Opposite charged ends of a neutral strand can join to create a neutral loop, so spin one bosons are represented by one neutral loop around one or two open strands and spin two bosons (gravitons) are represented by two neutral loops wrapped around one open strand. This immediately implies bosons can only have spin zero, one or two, because three loops leave no remaining open strands. Two charged strands of a single SM particle cannot interact electromagnetically because incoming charged strands in the cross channel of potential strand-strand electromagnetic interactions cannot combine to make the crossed channel three strand photon intermediate state needed to generate electromagnetic interactions between two charged strands. Neutral bosons involve neutral loops and non-identical neutral strands. 5

6 Denote strand (e/6, e/6) by A and strand ( e/6, e/6) by B. The model then allows for the spin states of all bosons that can exist as independent particles outside of color neutral hadrons. A linear superposition of configurations of three neutral strands with none wrapped around each other (AAA, AAB, ABB, and BBB) is identified with the scalar Higgs boson. A neutral loop wrapped around two open neutral strands (AA, AB or BB) provides three spin states of the Z 0 boson. A neutral loop threaded through another neutral loop, with an open strand (A or B) also going through the loop, gives two photon helicity states. Two neutral loops around an open strand (A or B) gives two (general relativitistic) graviton helicity states. Gluons cannot exist as independent particles outside of color neutral hadrons, and they are discussed in the next section on chromodynamics. W bosons are unique among SM bosons because they involve three charged open strands, either [( e/6, e/6), ( e/6, e/6) and ( e/6, e/6)] or [(e/6, e/6), (e/6, e/6) and (e/6, e/6)]. W bosons are represented by two bent open charged strands, forming an effective closed loop around another open strand. The state with an upper configuration of bent strands, with the strand open towards the negative x-axis above the strand open toward the positive x-axis, is identified as the state with s z = 1. The state with a lower configuration of bent strands, with the strand open towards the negative x-axis below the strand open toward the positive x-axis, is identified as the state with s z = 1. A superposition of states with upper and lower strand configurations is identified as the state with s z = 0. In the wrapped preon model, spin 1 particles are represented by one full 2 right- or left-handed spiral wrapped through an angle of 2π around one or two open strands. Clockwise or counterclockwise wrapping corresponds to spin up and spin down. The three possible ways this wrapping can occur allows three generations of spin 1 of fermions. The three configurations, shown 2 in Figure 1, are: a) two different open strands producing an effective full spiral wrap around a central open strand, b) a spiral wrapped around an open strand, accompanied by a spectator open strand, and c) a spiral wrapped around two open strands. Replacing various neutral strands with charged strands allows for three generations of SM fermions and no more. Neutrinos are linear superpositions of configurations with three neutral strands (AAA, AAB, ABB, and BBB), charge 1 quarks are linear superpositions of configurations with two neutral strands (AA, AB or BB), charge 2 quarks are 3 3 linear superpositions of configurations with one neutral strand (A or B) and unit charge fermions are configurations with all strands carrying the same 6

7 charge. Chromodynamics Each quark state has an odd strand out. The odd strand out is the neutral strand in charge 2 3 quarks and the charged strand in charge 1 3 quarks. Locating the odd strand out in one of the three different possible locations in the quark configurations for each generation provides for the color states of quantum chromodynamics. However, specifying a color charge for a particle requires an extra bit of information in addition to the six bits specifying the preon strands within that particle. So, to avoid conflict with the assumption that the N bits of information on the horizon comprise all information describing the universe, quarks must only occur as parts of color neutral composites (e.g., quark-antiquark pairs comprising mesons and color neutral three quark states comprising baryons) and gluons must not exist independently outside of hadrons. The gluon configuration in Figure 1 allows for eight different kinds of gluons confined within hadrons. The eight states are states with a neutral loop wrapped around one or the other of the open strands, an A or B strand inside the loop and an A or B strand as a spectator. Standard Model particle interactions SM boson and fermion interactions can be represented as preon interactions. For fermions interacting by neutral boson exchange, oppositely charged ends of strands of the incoming fermion-antifermion pair in the crossed channel join to produce three neutral stands propagating as the exchanged boson in the crossed channel. Later, any loops break and the three neutral stands split to create the strands of the outgoing fermion-antifermion pair in the crossed channel. Consider electron-electron interactions as an example. Electromagnetic interactions between electrons are generated by one photon exchange, with a one-photon intermediate state in the crossed channel electronpositron interaction. Incoming SM particles in the crossed channel include three strands ( e/6, e/6), ( e/6, e/6) and ( e/6, e/6) for the electron and three strands (e/6, e/6), (e/6, e/6) and (e/6, e/6) for the positron. Joining oppositely charged ends of pairs of incoming strands gives three neutral 7

8 strands to propagate as a photon in the crossed channel. Later, the photon loops break and the resulting three neutral open strands split to make the charged strands of the outgoing electron-positron pair in the crossed channel. Weak decay, exemplified by neutron decay to a proton, electron and antineutrino, occurs by converting a d quark to a u quark plus an electron and an anti-neutrino. The decay d u+e +v is equivalent to the crossed channel interaction d + u e + v. In the crossed channel, oppositely charged ends of pairs of incoming (e/6, e/6), (e/6, e/6) and ( e/6, e/6) strands for the d quark and (e/6, e/6), ( e/6, e/6) and ( e/6, e/6) strands for the u quark join to form three charged ( e/6, e/6), ( e/6, e/6) and ( e/6, e/6) strands propagating as a W weak boson in the crossed channel. Later, the three charged strands of the W boson split to make three charged strands of the outgoing electron and three neutral strands of the outgoing anti-neutrino in the crossed channel. This example shows that persistence of (e/6, e/6) and ( e/6, e/6) strands in a single SM particle is inconsistent with neutron decay mediated by a weak boson composed of three, and only three, preon strands. Preon dynamics Although the holographic principle implies local continuum theories like quantum field theory can only approximate the underlying finite-dimensional reality, interactions of three preon strands can be modeled by three-preon Bethe-Salpeter field-theoretic equations [4] involving only non-local threepreon forces. Instantaneous three-preon interactions (consistent with instantaneous transitions required in a non-local holographic quantum mechanical theory), and free particle propagators for the three strands, simplify these equations and reduce them to six-dimensional relativistically-covariant equations involving the six four-momenta p 1, p 2, p 3, p 1, p 2 and p 3 of incoming and outgoing states in three-strand interactions [4]. If the universe is closed, the number of bits of information in the universe is constant (because there is nowhere else to serve as a source or sink of information) and the number of bits of information in the universe is twice the (conserved) number of strands in the universe. So strand production thresholds are not an issue when dealing with the three-preon Bethe-Salpeter equations. The three-preon Bethe-Salpeter equations can be partial-wave analyzed for states of angular momentum L by the method of Omnes [5]. When the 8

9 instantaneous Born term V L (p 1, p 2, p 3 ; p 1, p 2, p 3 ) in the partial-wave Bethe- Salpeter equation has the non-local separable form V L (p 1, p 2, p 3 ; p 1, p 2, p 3 ) = g L( p 1 )g L ( p 2 )g L ( p 3 )g L ( p 1 )g L ( p 2 )g L ( p 3 ), the partial-wave three-preon Bethe-Salpeter equations can be solved algebraically (like the non-relativistic Lippman-Schwinger equations in Ref. 6), producing only one bound state configuration for each interaction. Therefore, SM particles can be identified as bound states of various non-local separable three strand interactions in three-preon partial-wave Bethe-Salpeter equations. A minimum of nine non-local separable interactions, one for each basic strand configuration, is needed to produce the structures in Figure 1. The required interactions are: V L=0 for three open strands of scalar bosons; three V L= 1 2 interactions (one each for the three spiral wrapped configurations of the three fermion generations); four V L=1 interactions (one for the three charged open strands in W bosons, one for the single loop around two open strands of the Z 0 boson, one for the single loop around a loop and an open strand for the photon, and one for the single loop around an open strand accompanied by a spectator open strand for the eight confined gluons); and one V L=2 interaction for the double loop around the open strand of the graviton. The resulting three-preon Bethe-Salpeter equations involving these non-local separable interactions can be seen as a continuum mathematics approximation to the underlying finite dimensional non-local theory of particle interactions. The far more difficult problem of developing a complete holographic theory of particle interactions based on this preon model involves at least three main steps: 1. Develop a finite-dimensional lattice gas model for the interaction of the bits of information on the horizon, representing SM particles moving in a Higgs boson sea. 2. Relate the lattice gas model to non-local nine strand vertices representing SM particle vertices within the universe. 3. Relate the non-local nine strand vertices to the known local field theoretic SM vertices. 9

10 Information and preon strands Consider the relation between the holographic information on the horizon and the distribution of preon strands within the universe. The wavefunction specifying the information on the horizon is the boundary condition on the wavefunction specifying the probability distribution of strands throughout the universe. The wavefunction for the probability of finding a strand anywhere in the universe is a solution to the Helmholtz wave equation in the universe. A closed universe with radius of curvature R can be defined by three coordinates χ, θ and φ, where the volume of the three-sphere (S 3 ) is π R 3 sin 2 χ dχ 0 π 0 sin θ dθ 2π [7]. The wavefunction for two bits of information on the horizon is the projection of the wavefunction for a strand in the volume of the universe on the horizon at R H = R sin θ. One solution to the Helmholz equation on the three-sphere [8] is the scalar spherical harmonic Q 2 10 = 12 π cosθ i sin χ. For the (i, j) bits of information associated with diametrically opposite pixels on the horizon, the cosθ i behavior of the wavefunction on the horizon determines the wavefunction 12 Ψ ( i, j) = C 3 Q 2 10 (i) = C 3 π cosθ i sin χ as the solution of the Helmholz wave equation describing the probability distribution within the universe for the strand associated with the (i, j) bits. 4 The constant C 3 = is determined by normalizing the wavefunction π 2 R 3 to one strand (associated with two of the N bits of information) within the universe. The probable number n of strands in a volume V within the universe is ( N ) 2 N/2 n(v ) = Ψ i,j (a i, a j ) dv, 2 V i,j=1 where all numbers a i and a j are either zero or one and together these N numbers encode all information about the distribution of matter and energy in the universe dφ

11 Conclusion For many years, particle physics employed the abstract concept of massive point particles. The concept is useful, despite the mathematical complications from infinite energy densities of massive point particles. Wrapped preon strands extend the idea of a massive point particle to the concept of a linear filament of energy with a charge at each end. If, as suspected, more than four dimensions are required for a successful theory involving all of the four forces in nature, the strands may represent energy associated with small extra dimensions. If this wrapped preon model reasonably represents reality, there is no room for supersymmetric partners or fourth generation SM fermions. Confirmation of the existence of supersymmetric particles or fourth generation SM fermions at the Large Hadron Collider, or anywhere else, will immediately falsify the model. References 1. Bilson-Thompson, S.O., A topological model of composite preons, hep-ph/ Mongan, T. R., A simple quantum cosmology, Gen. Rel. Grav. 33, , (2001), gr-qc/ Mongan, T. R., Holography and non-locality in a closed vacuum-dominated universe, Int. J. Theor. Phys. 46, , (2007), gr-qc/ Loering, U., Kretzschmar, K., Metsch, B.C., and Petry, H.R., Relativistic quark models of baryons with instantaneous forces, Eur. Phys. J. A (2001), hep-ph/ Omnes, R.L., Three-Body Scattering Amplitude. I. Separation of Angular Momentum Phys. Rev. 134, B1358, (1964) 6. Mongan, T.R. and Kaufman, L, Non-relativistic separable-potential quark model of the hadrons, Phys. Rev. D3, 1582, (1971) 7. J. Islam, An Introduction to Mathematical Cosmology, 2nd edition, Cambridge University Press, pages 43 and 44, Cambridge, U.K., (2002) 8. U. Gerlach and U. Sengupta, Phys. Rev. D18, 1773, (1978) 11

12 Figure 1: Three preon strand configurations for Standard Model particles. Strands are shown as tubes for ease of visualization. 12