Contents

A Novel Approach to Extra Dimensions


David J. Jackson





March 2, 2016


Abstract


Four-dimensional spacetime, together with a natural generalisation to extra dimensions, is obtained through an analysis of the structures and symmetries deriving from possible arithmetic expressions for one-dimensional time. On taking the infinitesimal limit this simple one-dimensional structure can be consistently equated with a homogeneous form of arbitrary dimension possessing both spacetime and more general symmetries. An extended 4-dimensional manifold, with the associated spacetime symmetry, provides a natural breaking mechanism for a higher-dimensional form and symmetry of time. It will be described how this symmetry breaking leads to a series of distinct properties of the Standard Model of particle physics, deriving directly from the natural mathematical development of the theory.

1 Introduction

In 4-dimensional spacetime an interval of proper time can be expressed locally in terms of the Minkowski metric for suitable local coordinates as:

(1)

with . A typical approach to extra dimensions would involve an extension to the range of the indices in the above quadratic expression, with the metric correspondingly replaced by an matrix for the -dimensional spacetime generalisation. An extended 4-dimensional spacetime manifold, with the local metric of equation 1, might be identified for example through the spontaneous compactification of the extra dimensions, in principle resulting in residual physical properties that might be observable in 4-dimensional spacetime.

For the theory described in this paper in place of generalising the right-hand side of equation 1 for a higher-dimensional spacetime structure we consider a generalisation of the overall expression as constrained by the time interval on the left-hand side. For example from the perspective of the linear one-dimensional flow of time it is equally permitted to write the cubic expression:

(2)

with each coefficient for for the -dimensional case. In the following section we describe how to express the general form of this type of generalisation. The means of identifying the 4-dimensional structure of equation 1 within this generalisation will be described in section 3 and will provide the means of breaking the full symmetry of expressions such as equation 2.

In sections 46 it will be described how natural mathematical extensions of this idea lead directly to the identification of physical properties such as fractional charges for quark states and a left-right asymmetry of a kind closely resembling the structure of the Standard Model. This development leads to the consideration of an symmetry in section 7 as a final stage in this progression that might in principle accommodate the full set of Standard Model properties.

2 Time and Spacetime

In this paper it is described how consideration of a multi-dimensional form of time and symmetry breaking over a 4-dimensional spacetime manifold leads directly to structures which exhibit a close resemblance to the Standard Model. We begin here by developing the underlying idea. A finite interval of time represented by the real number can be algebraically expressed in terms of other real numbers in an endless variety of ways, for example we can have and so on simply by employing the basic arithmetic structure of the real line.

The broad range of possible expressions for a finite interval in terms of an arbitrary number of variables {}, , will be constrained to a more restrictive structure in the limit of infinitesimally small temporal intervals. We first consider this limit for the trivial case with the flow of time expressed in terms of a single real variable only for which we have simply . This can symbolically be written as as we approach the limit of infinitesimal intervals. We then express the rate of change of with respect to in this limit as:

(3)

For the case with multiple real numbers representing the flow of time each will be associated with a corresponding rate of change with respect to time. For example, we may consider the propagation of time expressed for an infinitesimal interval as:

(4)
(5)

where (and with the conventional summation over repeated indices implied throughout this paper). Dividing by and taking the limit this can be written as or , which is invariant under the group, O(3), of orthogonal transformations in three dimensions applied to . The question is then how to express the general case for the composition and symmetries of a multi-dimensional set of velocities .

The infinitesimal elements of time can be written most generally, taking care to balance the order of the vanishing elements in each term, as:

(6)

Here the coefficients are each equal to or since we wish to express the purely in terms of simple arithmetic relations of the . In equation 6 each term divides into a separate portion of time:

(7)

where each term is the -root of a homogeneous polynomial of order in the {}. Taking each term in turn, dividing by the interval in each case and taking the limit we find:

(8)
(9)
(10)
(11)

where is a homogeneous polynomial of order in the components ; it can be considered as a map from the elements of a real -dimensional vector space onto the unit . Equation 11 is taken to express the general mathematical form of multi-dimensional temporal flow and it is the central equation of this paper. The symmetries of will be represented by groups acting on the vector space such that for all elements of the group and all vectors satisfying we have where represents the action of the group element on the vector .

Quadratic forms in general, including the 4-dimensional form:

(12)

that is for with the Minkowski metric , and with Lorentz symmetry, and the norm of an element of a division algebra ( or ), together with their symmetry groups, are expected to be particularly significant forms of . This is due to their close relation to Clifford algebras and Euclidean spatial geometry, describing the geometry of external space and spacetime.

Equation 12 is the special case of equation 10 in the form of the particular case of equation 1 described in the introduction. The generalisation to the cubic form in equation 2 is incorporated within the general expressions of equations 10 and 11. Other possible forms of include the determinants of matrices, which are homogeneous polynomials in the matrix elements.

With various different forms of progression in time to be considered, in general the subscript in the notation indicates collectively the vector space , the implied form and the corresponding symmetry group (respectively , and in the above example for ). The notation and , with a ‘hat’ above a kernel symbol, will denote the highest-dimensional form of time considered and its symmetry respectively.

Given a possible -dimensional form of progression in time, , the vector may be written as the ordered set of velocities:

(13)
(14)

the values of which are unchanged by a numerical translation of the real variables,

(15)

for any constant set , or for a subset of . Above we described a possible symmetry of with the action of a group mixing the numerical components , which represent elements of the flow of time . Here we have a further symmetry implicit in with respect to translations of the numerical variables as . That is, we also have trivially:

(16)

satisfying . The relation between the ‘translation symmetry’ of and the ‘rotation symmetry’, more generally denoted by the action for , is key to the development of the geometrical structure of the theory.

3 Extra Dimensions

We initially consider to be provisionally taken as the full symmetry group for the form , which in turn is the 10-dimensional extension of equation 12 (this is also equivalent to an extension for extra spacetime dimensions as described for equation 1). Here an extended base manifold arises through employing for four of the ten translational degrees of freedom of , in the manner described in equation 16. For this model we hence obtain the structures described in figure 1.

Figure 1: (a) The full symmetry group over the base space (b) broken to the internal symmetry SO(6) with external subgroup acting on the tangent space . (An underline, as for , or an overline, as for , may be used for internal and external objects respectively, in case of ambiguity).

This figure also represents the manner in which the identification of an extended 4-dimensional background manifold breaks the symmetry of the higher-dimensional form of time . However while quadratic spacetime forms such as are included as a possible structure of equation 11 more generally cubic or higher-order polynomial forms of time are also permitted (as initially described for equation 2).

We approach this generalisation via the group as the two-to-one cover of , which may be exhibited by mapping a Lorentz vector into the space of complex Hermitian matrices as:

(17)

where denotes the identity matrix together with the three Pauli matrices , that is , , . While the fundamental representation of acts on the space , the group action for elements on the space provides another representation given by:

(18)

This maps onto a new complex Hermitian matrix with the same determinant; hence mapping the components according to a Lorentz transformation of the real 4-vector . This action expresses the symmetry of of equation 12 in a manner that naturally extends, consistent with equation 11, to an symmetry of the cubic polynomial form with . For this latter case, in identifying the base space through the translation symmetry of subspace of external vectors , the symmetry is broken to . The action of the external symmetry on the full space may then be considered. The matrices can be embedded in matrices acting on as:

(19)

This combines the vector representation of on and the spinor representation on (taken to be left-handed), together with the scalar denoted (in line with the notation used for equation 20 below), in a single symmetry transformation which preserves .

Hence by considering a higher-dimensional cubic form of time we have identified components of the ‘extra dimensions’ transforming as a spinor, namely the of equation 19, which also transforms non-trivially under the internal action identified in the symmetry breaking. The is in contrast to Kaluza-Klein theories in which a 4-vector object , exhibiting properties of gauge field, can be identified in the extra components of a higher-dimensional metric tensor, as for example in the metric case as originally formulated in the 1920s [1, 2]. In the following section we consider a further natural extension of equation 19 for a higher-dimensional form of for the present theory.

4 Symmetry on

A natural generalisation from the space , underlying the vector transformed in equation 19 as a symmetry time, is obtained by augmenting the complex numbers to the largest division algebra, namely the octonions  [3]. The vector space obtained corresponds to the set of Hermitian matrices over the octonions with elements which can be written as (in this paper we closely follow [4], [5] chapters 3 and 4, together with [6, 7, 8], for all details of the structure, and generally adopt the notation therein):

(20)

with (here the component labels are chosen to conform with the notation in the main references, and here is of course not the dimension of any space), and denotes the octonion conjugate of reversing the sign of the 7-dimensional imaginary part. In general an octonion can be described by eight real parameters and written as:

(21)

The seven imaginary units in this basis , with , are mutually anticommuting, with etc., with their full algebraic composition described in [5, 6]. While the octonions are non-associative they compose the largest ‘normed division algebra’ [3]. In equation 20 and have the structure of an octonionic vector and spinor respectively. Hence the vector space is 27-dimensional over the real numbers. It is a space with particularly rich symmetry properties largely owing to the nature of the 8-dimensional octonion subspaces [3].

As for the space a cubic norm, or determinant for , can be defined on the space taking the form:

(22)
(23)

where the 10-dimensional Lorentz inner product with , in the first line, together with equation 20 can be used to derive the second line in which the cubic composition of components, consistent with the homogeneous form of equation 11 (and with equation 2 as a particular cubic form), is explicitly seen. The symmetry of the form , that is the symmetry leaving invariant, is a real form of as we briefly review here. We closely follow references [4, 5, 6, 7, 8] within which, in particular, the means of accommodating, and employing, the non-associative property of the octonions is described in detail.

As a generalisation from the Lorentz transformations of equation 19 a set of matrix actions with can be embedded in the upper-left corner of matrices to obtain a conjugation action for the case with:

(24)

This expression contains the vector , spinor and scalar representations of , each transforming in the appropriate way with the form of the action determined correspondingly. These transformations respect the vector and spinor block structure described in equation 20.

With three natural ways to embed the transformations the action depicted in equation 24 is denoted as ‘type 1’ The form of matrices transforming under the type 1 and actions is compatible with the isomorphism of vector spaces ([3] p.30):

(27)

The three parts of this decomposition are respectively the scalar, vector and spinor representations of the 10-dimensional spacetime symmetry group , for which the covering group is . The object , from equation 20, corresponds to the Majorana-Weyl spinor representation, also denoted as .

With the group being 45-dimensional and with three distinct types of embedding this implies to a total of determinant preserving actions on the space . The collective symmetry of these actions is described in terms of vector fields on the tangent space to :

(28)

where parametrises a given action. By studying the linear dependences of the resulting 135 vector fields it can be shown [5, 6, 7, 8] how a linearly independent basis of 78 actions emerges which fully describes as the determinant preserving transformations on . The entire group is then described in terms of the actions of matrices on the space , with the preferred basis for the Lie algebra represented on reproduced below in table 1.

Category 1: Boosts #
Category 2: Rotations
Category 3: Transverse Rotations
Total Generators 78
Table 1: The complete basis for the Lie algebra of , in terms of tangent vector fields on , reproduced from ([5] p.177, table A.1). The superscripts denote the ‘type’. The subscript denotes any of the seven imaginary octonion units .

Here all 78 generators are explicitly determined and listed in tables 6 and 7 in the appendix of this paper, for the category and 3 transformations respectively, as tangent vector fields which, from equation 20, are of the form:

(29)

The Lie algebra commutator, which determines the structure constants of the Lie algebra, for any two elements is defined through the action of the respective one-parameter subgroups and at any point :

(30)

The full Lie algebra commutation table is available in [5], for which the full set of independent entries were found by computer program. For this paper sign and other conventions have been tuned both for internal consistency and for consistency with the entries of the full algebra table [5]. (In this paper ‘’ may refer to either the Lie group or the Lie algebra, depending on the context, with a similar convention for the other exceptional Lie groups).

5 Symmetry Breaking Structure

We can identify the Lorentz 4-vector in the upper left-hand matrix embedded within the larger matrices in , as was the case for in equation 19. However here the preferred subspace basis is taken to be for , with one of the imaginary octonion units introduced in equation 21, as indicated in equation 32. The relation with is preserved under operations of representing the Lorentz group upon this space as:

(31)
(32)

with , and with ‘’ describing the identity transformation in the trivial 1-dimensional representation of this group, acting upon the components of of equation 20. This action preserves the value of det, as it is simply the transformation of equation 18, as well as leaving invariant. In equation 31 denotes the 6-dimensional imaginary part of of equation 20, that is excluding the real and imaginary components of which are associated with the external 4-vector . Since the actions, based on the complex subspace, are embedded in the ‘type 1’ location this group will be denoted . This describes the Lorentz subgroup for 4-dimensional spacetime generated by the subset of Lie algebra elements in from table 1:

(33)

With the group action of on in equation 31 embedded within the type 1 group action of on the same space as displayed in equation 24 we can write:

(34)

where the first ‘’ strictly applies at the Lie algebra level. This shows explicitly how the action of the Lorentz group may be embedded within the higher symmetry group acting on the space . The direct physical interpretation of the former symmetry as an action on the external spacetime manifold (as originally depicted in figure 1(b)) provides a direct source for the breakdown of the latter symmetry.

The two-sided action on in equation 31 only transforms the real diagonal entries and together with the and components of . The six components of remain invariant as may be deduced from the form of the six generators in table 6. (This is equivalent to the invariance of under for the model of figure 1). Of the additional 17 components in the real diagonal entry is also invariant, as is clear from equation 31, while all 16 components of transform non-trivially under the one-sided action.

The spinor will denote the components of and in , that is restricted to the complex subspace. By comparison with equation 19 this object transforms as a left-handed Weyl spinor under the action in equation 31. Here we take this action on to define the left-handed spinor representation.

Considering the three quaternionic subspaces with the base units , and it can be seen through explicit calculation that the original full octonionic spinor , with 16 real components, reduces to a total of four left-handed Weyl spinors under the action of , namely:

(35)

with the respective components of each transforming in an identical manner. Overall the decomposition of the representation of under the subgroup of equation 27 further reduces under the subgroup as summarised in table 2.

Components
  scalar (0,0) scalar
  vector
  spinor spinors
Table 2: The further decomposition of the representation of of equation 27 under the subgroup of external 4-dimensional spacetime symmetry actions of equation 31, and the corresponding components of transformed.

As a preliminary definition, and in contrast to the external symmetry, the internal symmetry will be obtained from the set of all actions on which leave the four components for any in equation 32 (that is, of equation 31) invariant. These components, including associated with the imaginary unit of , can also be expressed in the combination with respect to the parametrisation of equations 20 and 21. The corresponding symmetry group is complementary to the actions of and will be denoted as the stability group of all vectors . By inspection from tables 6 and 7, for the 78 elements in the preferred basis for the Lie algebra of defined on the space , the group is generated by the 31 elements listed in table 3.

Category 1 and 2: Boosts and Rotations #
2
14
Category 3: Transverse Rotations
,  ,   9
6
Total 31
Table 3: The Lie algebra generators of the 31 dimensional group . The subscript denotes any of the seven imaginary octonion units unless stated otherwise.

Of particular interest is the subgroup associated with the 8 generators (as also described in [5] pp.115 and 136, following [9] in relation to the colour symmetry). This is defined in terms of the transverse rotations acting on the octonion space alone as the subgroup of the octonion automorphism group that leaves one imaginary unit, here , invariant. The corresponding Lie algebra , described by the set of 8 generators , as transformations of on the full space , act on each of the octonion elements in the same way (as described in the table 7 caption) leaving invariant the complex subspaces, and as elements of table 3 identified within may be provisionally associated with the colour su(3) of the Standard Model for the present theory. This algebra is also independent of in terms of the Lie bracket composition, that is for all and , and hence we have the semi-simple subgroup:

(36)

The algebra elements, as transformations on the space mix the components of the spinor . For example the tangent vector field on the components of , obtained from table 7, are:

(37)

The components here have been ordered to match those of the four left-handed Weyl spinors of equation 35. The fact that each real component of transforms in the same way as the corresponding component of is expected since acts on each of in precisely the same way. However, it is also noted that the action in equation 37 respects the 4-way spinor decomposition.

The extraction of the components of a spinor into a matrix of real numbers will be denoted by . For example, from equation 35 the spinor can be mapped to the matrix of real numbers (with components ordered to match those of the spinor under transformations, as described for equation 35). Using this notation, and with representing the zero matrix, the tangent vectors of all eight generators of SU(3), including from equation 37, on the spinor space are listed in table 4 alongside the actions of the Gell-Mann matrices , using the correspondence of ([5] p.137, table 4.5), on the vectors . On the left-hand side the elements are already expressed as tangent vectors, while on the right-hand side the tangents are obtained by matrix multiplication of the into .