**THE
REGGE CALCULUS OF THE FUNDAMENTAL PARTICLES**

**QUANTIZED
DEFECTS IN DISCRETE SPACETIME**

Copyright 1988-2015 John Bruce Davies

This original research was awarded an honourable mention in the 1997 Gravity Research Foundation essay competition.

**Abstract**

Feynman, as well as numerous other famous physicists, has postulated that spacetime could be discrete rather than a continuum at the quantum level. However, this hypothesis has been only applied at the Planck level where Wheeler, Hawking and others argue for a spacetime foam. We, in the following, will argue and demonstrate that such a discrete spacetime is valid at the fundamental particle level as well. A fundamental tenet of the quantum theory is the Uncertainty Principle. We shall show that a discrete analogue of the Uncertainty Principle is a direct consequence of our approach.

A discrete space-time, governed by Regge calculus, is used to model the fundamental particles and their interaction. In this calculus, curvature is distributed over the 2-simplex defect with energy-momentum concentrated on the legs of the 2-simplex. We show that a baryon, composed of 3 quarks, is equivalent to the triangular 2-simplex, where the quarks compose the legs. Mesons and leptons correspond to collapsed line and point 2-simplices, respectively, where curvature and Action remain well-behaved through the collapse. There are 6 invariants of the 2-simplex that we argue correspond to the 6 quark flavors. The 3 on-diagonal invariants have inner products of 2/3 while the 3 off-diagonal invariants have -1/3 inner products. The closure condition on these invariants yields confinement of the quarks.

**Introduction**

The standard model in particle physics has been constructed mainly to satisfy the observed symmetries and related phenomenology. In this phenomological approach, extreme difficulty is found in relating the quantum structure of matter to the gravitational properties of the space-time continuum. To avoid these problems, we argue for a discrete space-time to be assumed at the quantum level. We show that quantized defects in such a discrete structure correspond to that of the massive fundamental particles. This reverses the phenomenological approach of the standard model and instead obtains particle structure from a basic space-time model.

In the standard model, certain arbitrary choices of particle structure and parameters are necessitated. Such phenomenology includes the gauge groups and their representations as well as the quantum numbers and family structure of the quarks and leptons. However, our discrete approach explains these phenomena using the basic properties of a self-consistent defect spacetime governed by Regge Calculus, as developed by Regge (1961) and explained in the ‘Gravitation’ magnus opus of Misner, Thorne and Wheeler (1973).

By incorporating both asymmetric curvature and torsion, we have shown, in the previous chapters, that linearised Bianchi Identities govern self-consistent defects in a spacetime continuum, Davies (1988). Using these equations, a model of particle pair production shows that particle density distributions and associated fields are defects in space-time. This geometric approach can be applied to a discrete space-time with defects where, for simplicity, only symmetric curvature will be assumed to affect the metric. Defects in a discrete space-time are incorporated in the particular lattice structure governed by Regge Calculus. It has been shown that symmetric curvature is located on 2-simplexes in such a discrete space-time. Previously, Regge calculus has been used to numerically model a continuum and its evolution, Wheeler (1964). However, it is usually argued, c.f. Hawking (1978), that Regge calculus will only apply at the Planck scale governing the postulated space-time foam. We show that it can also be used to describe phenomena at the quantum particle level.

The quantum theory governing particles has, as its basic foundation, that Action must be measured in discrete units of "h", Planck’s Constant. It is thus necessary that Action should be well-behaved when space-time is discretized from the continuum to the quantum lattice. The power of Regge calculus is that Action is well-defined and equivalent when taken in the continuum and the discrete limits, Regge (1961). Regge calculus decomposes space-time into a net of simplices, where Curvature is localised on the 2-simplex in the lattice. Energy-momentum resides in the legs, or 4-vectors, of the 2-simplex.

We postulate that a quantised 2-simplex is equivalent to the fundamental particle in this discrete space-time. The constituent quarks then correspond to the legs of the 2-simplex which carry the energy-momentum. The triangular 2-simplex corresponds to a baryon composed of three quarks. Such a triangular system can collapse to line and point simplexes with Action remaining well-behaved and Curvature remaining finite, Hawking (1978). The collapsed line 2-simplex thus corresponds to mesons while the point 2-simplex represents leptons.

The symmetry SU(3) of the baryon is inherent in the triangular nature of the 2-simplex. Examination of the affine coordinates of the 2-simplex shows that they have 6 invariants, 3 with 2/3 inner product and 3 with -1/3 inner product, as observed for quarks. Other phenomenology of the fundamental particles, such as quark confinement, are shown to be a consequence of the properties of these invariants of this discrete space-time 2-simplex. We show that Heisenberg’s Uncertainty Principle is inherent in this approach, a result that Hawking, in his popular book, maintains is necessary for any unification of gravity and quantum mechanics.

**Discrete
Spacetime**

Rather
than fill space-time with a grid of points, Regge calculus divides it
into a net of 0,1,2,3 and 4-simplexes. Fundamental quantities and
operations of the continuum theory have their discrete simplectic
analogs as singular instances of the continuum case[6]. The net is
endowed with metrical character by assigning to each leg [ij] of the
net a length l(ij). For a 4-simplex there are ten lengths l(ij) which
determine the ten independent components of the symmetric metric g_{μν}
.

This flat metric is defined for the interior of each 4-simplex cell but not for the boundaries of the interfaces between the cells. For 4-dimensional space-time, these boundaries are 2-simplexes and it may not be possible to find a coordinate system to cover smoothly all the cells that meet there. The deviation from flatness at this 2-simplex is characterised by the "defect angle" Φ . It is at these 2-simplexes that curvature is concentrated corresponding to the rotational or conical defect.

The continuum expression for the Action is:

(1) S = -1/2 ∫ R.dV

where the Ricci curvature is defined as:

(2)
R = g^{μα}^{
}g^{γβ}
R_{μναβ}

The only contribution to R is from the neighborhood of the 2-simplex defect. This expression for Action is translated into the discrete lattice and becomes, Regge (1961):

(3)
S_{g}
= Σ
Φ(ijk).A(ijk)

where the sum is taken over all 2-simplex defects and A is the area of the defect with angle Φ . The area A is given by:

(4)
A(ijk) = 1/4( l^{4}(ij)
+ l^{4}(jk)
+ l^{4}(ki)
- 2(l^{2}(ij)l^{2}(jk)+l^{2}(jk)l^{2}(ki)+l^{2}(ki)l^{2}(ij)))^{1/2}

The discrete curvature is obtained through differentiation of the Action with respect to the square of the lengths:

(5)
G(ij) = δS_{g}/δl^{2}(ij)

and is given by:

(6)
G(ij) = 1/16 Σ_{k
}Θ(ijk).σ(ijk).Φ(ijk).(l^{2}(ij)
- l^{2}(jk)
- l^{2}(ki))/A(ijk)

in which Θ(ijk) is 1 if i,j,k are the vertices of a 2-simplex defect but is zero otherwise, Sorkin (1974), and where σ(ijk) is the sign of the term under the square-root on the right hand side of eqn. (4).

As in the continuum case, when matter is present, the Einstein Equations become:

(7) G(ij) = T(ij)

and since T(ij) represents mass energy-momentum, then even though curvature is diffused throughout the 2-simplex defect, energy-momentum is concentrated along the legs of the 2-simplex net, Meisner, Thorne and Wheeler (1973).

**Invariants
of the 2-Simplex**

In this discrete space-time model of the elementary particles, we shall examine the properties of the invariants of these 2-simplices. We shall use affine coordinates, which, for tensors on an n-simplex, are relative to the barycenter of the simplex. In such an affine space, a vector u is defined by its end points, P and Q, where the locations are given by:

(8)
P
= Σ
p^{j}
. u_{j}
; Q
= Σ
q^{j}
. u_{j}

where
p,q are the coordinates of the points in affine space. The vector PQ
(**x**)
is the difference of these locations in the affine space and is given
by:

(9)
x^{j}
= p^{j}
- q^{j}

For the n-simplex with normalised affine coordinates, it has been shown, Sorkin (1974), that such affine systems are closed, i.e.:

(10)
Σ_{j}^{n}
x^{j}
= 0

The
fundamental basis set of vectors (**e**)
relative to the barycenter of the n+1 vectors comprising the
n-simplex is:

(11)
e_{i}
= u_{i}
– 1/(n+1) Σ_{k=0}^{n}
u_{k}

The invariants of this basis set are given by the inner product of the basis vector and its dual:

(12)
e^{j
}.
e_{k}
= δ^{j}_{k}
- 1/(n+1)

where δ is the usual delta function equal to 1 when j=k and 0 otherwise. The closure equation (10) becomes:

(13)
Σ_{k}
e_{k}
= 0

and similarly for its dual.

Putting n=2 into equation (12), it can be seen that the 2-simplex has 3 on-diagonal invariants:

(14)
e^{j
}.
e_{j}
= 2/3

each one having an inner product equal to 2/3. This equation also yields 3 off-diagonal invariants, where j is not equal to k, with a 1/3 inner product:

(15)
e^{j
}.
e_{k}
= 1/3

**Quantisation
and Fundamental Particles**

Quantum mechanics in curved and continuous space-time has problems due to perturbation convergence which is automatically avoided by use of a discrete lattice. A number of methods of quantising Regge calculus has been developed based on path integral, Hawking (1978), and canonical methods, Rocek and Williams (1981).

However, we utilise the fact that the fundamental basis of quantum mechanics is that Action is measured in units of h, Plancks' Constant. The Regge lattice is important in this context because the continuous limit of the discrete Action is equal to the continuum Action, Regge (1961), Sorkin (1975). Thus, in the discrete relation (3), we argue that discrete Action is to be measured in units of h.

In a quantum Regge lattice, we therefore argue that as energy-momentum resides on the legs of the particular 2-simplex, then this 2-simplex and its legs must represent fundamental massive particles. Thus the intimate relationship between the gravitational metric structure of space-time and its constituent quantised particles are related at this discrete level. The fact that only pairs of 2-simplices may be formed with equal but opposite properties, MTW (1973), is equivalent to pair-production of a particle and its anti-particle. Such a model of pair-production as equivalent to formation of defects in a continuum space-time was shown to yield the continuum equations governing particles intermediated by the electroweak forces, Davies (1988).

In the standard model, baryons are composed of three quarks which carry the energy-momentum of the fundamental particle. Such a structure for the baryon is readily observed in our 2-simplex triangle geometry. The quarks are 4-vectors and correspond to the legs of the 2-simplex as we have previously shown in equation (7) that the energy-momentum in Regge space-time is concentrated along these legs. Mesons, in the standard model, are composed of two quarks whereas Leptons are considered to be point particles.

Hawking (1978) has examined the effect of the collapse of a triangular 2-simplex to a line and point simplex and has shown that Action remains well-defined through the collapse. Taking the relation (6) for curvature on the legs of a 2-simplex, it is seen that in the limit of one leg approaching a zero length that curvature remains finite. Similarly, in the limit of all lengths of the legs approaching zero, the curvature will also remain finite and well-behaved. By considering a collapsed line 2-simplex as a meson and a totally collapsed point 2-simplex as a lepton, we can understand the structure of the fundamental massive particles through the geometry of these discrete 2-simplices.

We postulate that the 6 invariants of the 2-simplex corresponds to the 6 quark flavors with quantum numbers 2/3 and 1/3. The closure equation (13) implies that the vector addition of these invariants is zero in this 2-simplex. This is equivalent to color force confinement of the quarks which is considered to be similar to vector addition. The color gauge group SU(3) is readily comprehended from the triangular geometry of the 2-simplex.

**Uncertainty
Principle**

For a single 2-simplex the Action is proportional to Area, so that an incremental change in Action necessitates a corresponding incremental change in Area for constant deficit angle. For an arbitrary quantised 2-simplex this entails that:

(16) δ l(ij) . δ l(jk) >= h

i.e. the increment of area must be greater than or equal to h, Planck's Constant. This is the discrete analogue of Heisenberg's Uncertainty Relation. In Regge calculus, energy-momentum resides along the legs of the 2-simplex, while curvature is spread over the enclosed area. The above inequality, proves that the basis of quantum mechanics in continua is also a fundamental tenet of quantum mechanics of discrete spacetime. Thus, through this discretization, we are finally able to unite quantum mechanics, through energy-momentum of particles, with curvature of the particles’ matter in spacetime.

**References**

Davies, J.B., 1988, ‘New Curvature-Torsion Relations through Decomposition of the Bianchi Identities’, Foundation of Physics, 18, 5, 563.

Davies, J.B., 1997, ‘The Regge Calculus of the Fundamental Particles’, Honorable mention, Gravity Research Foundation essay competition.

Hawking, S.W., 1978, Nuclear Physics B144, 349.

Misner, C.W., K.S. Thorne & J.A. Wheeler, 1973, ‘Gravitation’, Freeman Press, New York.

T. Regge, 1961, Nuovo Cimento 19, 558.

M. Rocek and Williams, R.M., 1981, Physics Letters 104B, 31.

Schouten, J.A., 1954, ‘Ricci-Calculus’, 2nd edn., Springer-Verlag Press, Berlin.

Sorkin, R., 1974, Ph.D. Thesis, California Institute of Technology, CA.

Sorkin, R., 1975, Physical Review D 12, 385.

Wheeler, J.A., 1964, in ‘Relativity, Groups and Topology’, Gordon & Breach Press, New York.