The Bekenstein-Sanders [1] [2] tensor-vector-scalar theory of gravitation (TeVeS) has been shown to account for the galactic rotation curves, lensing, and other cosmological phenomena (see review of Skordis [3] ) without the significant presence of dark matter¹.

It has recently been shown [6] , that there is an invariant Hamiltonian formalism for the TeVeS theory, achieved by a conformal transformation, for which the essential Bekenstein-Sanders field B μ , satisfying B μ B μ = − 1 , emerges as a gauge field (see also [7] for the many body case). Since the normalization condition B μ B μ = − 1 must be maintained under gauge transformations, it is necessary that the field B μ be non-Abelian, similar to a Yang-Millls [8] field.

The interesting possibility that the field B μ can be represented as a quaternionic field is investigated in this paper. This possibility would imply that the quantum mechanical wave functions for which B μ is the gauge field are also quaternionic, as discussed by [9] [10] .

In the following we will be working in the framework of the embedding of the relativistic quantum theory [11] in the curved space of Einstein’s general relativity [12] [13] . The vectors and tensors we shall discuss, and local partial derivatives are well-defined in the local tangent space at each point.

The dynamics of such quaternionic wave functions has been discussed by Adler [14] using trace dynamics, thereby opening up a possibly fruitful field relating quaternionic quantum mechanics and general relativity.

As discussed in [9] , the quaternionic wave function ψ ( x ) may undergo left and right gauge transformations

ψ → ψ _ = ω ψ ω ′ , (2.1)

with ω ω * = ω ′ ω ′ * = 1 , where { ∗ } is the quaternion conjugate, for complex units { e i } , i = 1 , 2 , 3 and e 1 e 2 e 3 = − 1 , e i 2 = − 1 and cyclic, ( e i e j ) * = e j e i , i ≠ j . The prime indicates the left gauge.

The covariant derivative [9] is defined by ( ∂ μ ≡ ∂ ∂ x μ , and indices are raised

and lowered by the Minkowski metric η ν μ = { − 1, + 1, + 1, + 1 } )

D μ ψ = ∂ μ ψ + B μ ψ − ψ B ′ μ , (2.3)

with B μ * = − B μ , B ′ μ * = − B ′ μ . Under gauge transformations of the form (2.1),

B μ → ω B μ ω * + ω ∂ μ ω * B ′ μ → ω ′ B ′ μ ω ′ * + ω ′ ∂ μ ω ′ * (2.4)

Differentiating ω ω * = 1 , we see that

ω ∂ μ ω * = − ∂ μ ω ω * = − ( ω ∂ μ ω * ) * (2.5)

so the additional terms in Equations (2.4) are pure quaternion imaginary.

Under the general gauge transformation

D _ μ ψ _ = ∂ μ ( ω ψ ω ′ * ) + ( ω B μ ω * − ∂ μ ω ω * ) ω ψ ω ′ *       − ( ω ψ ω ′ * ) ( ω ′ B ′ μ ω ′ * + ω ′ ∂ μ ω ′ * ) = ∂ μ ( ω ψ ω ′ * ) + ( ω B μ ψ ω ′ * − ∂ μ ω ψ ω ′ * ) − ω ψ B ′ μ ω ′ * − ω ψ ∂ μ ω ′ * = ω ∂ μ ψ ω ′ * + ω B μ ψ ω ′ * − ω ψ B ′ μ ω ′ * = ω D μ ψ ω ′ * , (2.6)

showing that the covariant derivative of ψ transforms under gauge transformations in the same way as ψ [9] .

We will be primarily interested in the left gauge in the following (because of the structure of the quantum quaternionic scalar product, as we shall see) but a similar argument is effective for the right sided gauge as well.

It is essential for the Bekenstein-Sanders results that, as mentioned above, under gauge transformations, the relation

B μ B μ = − 1, (2.7)

requires that the gauge field be non-Abelian. The proof that there is a class of gauge transformations which preserves (2.7) can most easily be carried out for infinitesimal gauge transformations. With the help of (2.5), we may write the transform of (2.7) as

B μ B μ → ( ω B μ ω * + ω ∂ μ ω * ) ( ω B μ ω * − ∂ μ ω ω * ) = ω B μ B μ ω * − ∂ μ ω B μ ω * + ω B μ ∂ μ ω * − ∂ μ ω ∂ μ ω * (2.8)

The first term on the right provides the necessary −1, so we must show that

− ∂ μ ω B μ ω * + ω B μ ∂ μ ω * − ∂ μ ω ∂ μ ω * = 0. (2.9)

Moreover, since

( ∂ μ ω B μ ω * ) * = − ω B μ ∂ μ ω * (2.10)

we have, from (2.9), the requirement

2 R e ∂ μ ω B μ ω * − ∂ μ ω ∂ μ ω * = 0. (2.11)

We now show that there exist solutions for this nonlinear relation by studying infinitesimal local gauge transformations of the form ( ε real and small), for a neighborhood of some x μ ,

ω = 1 + ε v , (2.12)

with v pure quaternion imaginary, so that

ω ω * = ( 1 + ε v ) ( 1 − ε v ) = 1 + O ( ε 2 ) .

Now, substituting (2.12) into (2.11), one finds, to O ( ε 2 ) , that we must have

R e B μ ∂ μ v = 0. (2.13)

Since B μ is timelike, there is a (local) Lorentz frame for which only its time component is non-zero; in this frame,

R e B 0 ∂ 0 v = 0. (2.14)

For

B 0 = e 1 b 1 + e 2 b 2 + e 3 b 3 ∂ 0 v = e 1 ∂ 0 v 1 + e 2 ∂ 0 v 2 + e 3 ∂ 0 v 3 (2.15)

from which it follows that

R e B 0 ∂ 0 v = − ∑ i = 1 3 b i ∂ 0 v i . (2.16)

It is therefore necessary and sufficient (by successive infinitesimal transformations), that in this local frame, the quaternionic parts of the time deivative of the infinitesimal gauge transformation be orthogonal to the quaternionic vector part of B 0 . Since (2.12) is invariant under local Lorentz transformations, this result implies that (2.13) must be valid as well, at any point in the manifold, implying that there is a class of gauges that leaves B μ B μ = − 1 .²

Although the quaternionic wave function has the property that it can carry left or right gauge trnsformations, it will be convenient (and sufficient for our present purposes) to use the left gauge.³

Consider a local single particle gauged Hamiltonian of the form

K = 1 2 m g μ ν ( x ) ( p μ − ε B μ ( x ) ) ( p ν − ε B ν ( x ) ) + Φ ( x ) , (3.1)

where Φ is a (real-valued) world scalar field, K is quaternion real, g μ ν is the (real-valued) Einstein metric, p μ is quaternion imaginary (discussed in [9] ), and B μ is the quaternionic Bekenstein-Sanders field. We define, as in [6] , a conformally modified metric

g ^ μ ν = g μ ν K K − Φ (3.2)

Since g μ ν is real-valued, we may cancel K from both sides, and multiply by ( K − Φ ) to show the equivalence between (3.2) and (3.1).

Defining, as in [1] [2] [6] ,

K K − Φ ≡ e − 2 ϕ , (3.3)

the Hamiltonian

K K = 1 2 m g ˜ μ ν p μ p ν , (3.4)

for [1] [2]

g ˜ μ ν = e − 2 ϕ ( g μ ν + B μ B ν ) − e 2 ϕ B μ B ν , (3.5)

K K = e − 2 ϕ g μ ν p μ p ν − 2 sinh 2 ϕ B μ B ν (3.6)

is equivalent to (3.1), generating the same equations of motion [11] [15] .

We argue here that for p μ = q ∂ μ , with q imaginary quaternionic [9] ,

[ q , ω ] = 0 [ B μ , q ] = 0 (3.7)

The first of (3.7) is implied by the requirement

( p μ − B _ μ ) ψ _ = ( q ∂ μ − B _ μ ) ω ψ = ω ( q ∂ μ − B μ ) ψ , (3.8)

or

q ( ∂ μ ω ) ψ + q ω ∂ μ ψ − B _ μ ω ψ = ω q ∂ μ ψ − B μ ψ . (3.9)

The gauge condition

B _ μ = ( q ∂ μ ω ) ω − 1 + ω B μ ω − 1 (3.10)

follows if [ q , ω ] = 0 , so that the ∂ μ ψ term cancels on both sides.

Furthermore, since q is constant, and we take it to commute with ω,

[ B _ μ , q ] = ω [ B μ , q ] ω − 1 ; (3.11)

if we start with ( B μ ) i n i t i a l = 0 , it follows from (3.10) that a first gauge step to ( B _ μ ) n e x t will still commute with q. This condition is maintained for any sequence of ω’s (commuting with q), and, therefore, for any B μ constructed in this way.

We may now define a Kaluza-Klein metric [15]

g A B = ( g μ ν B μ B μ g 55 ) . (3.12)

If we take [6]

p 5 = − p μ B μ g 55 ( 1 ± 1 − 2 g 55 sinh 2 ϕ ) , (3.13)

then

K K = 1 2 m g A B p A p B . (3.14)

Wesson [16] and Kaluza [15] chose g 55 − c o n s t ; in our context, it may be taken to be zero.

We have discussed a quaternionic formulation of the Bekenstein-Sanders [1] [2] TeVeS gravitational theory. It was shown in [6] that this theory can be derived by a conformal transformation from a Hamiltonian form on a curved space [13] , for which the Bekenstein-Sanders vector field B μ is a non-Abelian gauge field. We give here a quaternionic formulation suggested by this structure. We proved for this quaternionic formulation (as well as provided a missing proof for the Yang Mills form [6] ) that there is a set of gauge transformations that preserves the Bekenstein-Sanders condition B μ B μ = − 1 . It has been shown [7] that one can construct a theory for N ≥ 2 particles in such a TeVeS theory, suggesting that a rigorous statistical mechanics could be developed (see also Giordino et al. cited in [3] ).

Since the wave functions in the Hilbert space, carrying the non-Abelian quaternionic guage, are quaternionic, as dynamical variables they may satisfy the trace dynamics developed by Adler [14] , opening a subject for future research, relating quaternionic quantum mechanics to general relativity.

The author declares no conflicts of interest regarding the publication of this paper.

Horwitz, L. (2023) Quaternionic Bekenstein-Sanders Guage Fields for TeVeS. Journal of Modern Physics, 14, 1203-1210. https://doi.org/10.4236/jmp.2023.148066

For standard Yang-Mills theory [6] (result stated but not proved there), under gauge transformation,

B _ μ B _ μ = ( ω B μ ω * − i ε ∂ ω ∂ x μ ω * ) ( ω B μ ω * − i ε ∂ ω ∂ x μ ω * ) = ω B μ B μ ω * − i ε [ ∂ ω ∂ x μ B μ ω * + ω B μ ω * ∂ ω x μ ω * ] − 1 ε 2 ∂ ω ∂ x μ ω * ∂ ω ∂ x μ ω * . (A.1)

Now, differentiating ω * ω = 1 (as above),

ω * ∂ ω ∂ x μ ω * = − ∂ ω * ∂ x μ , (A.2)

we find from (A.1) that

B _ μ B _ μ = − 1 − i ε [ ∂ ω ∂ x μ B μ ω * − ω B μ ∂ ω * ∂ x μ ] + 1 ε 2 ∂ ω ∂ x μ ∂ ω * ∂ x μ (A.3)

Now,

( − ω B μ ∂ ω * ∂ x μ ) * = + ∂ ω ∂ x μ B μ ω * (A.4)

so that, to maintain the relation B _ μ B _ μ = − 1 , we must have

2 R e [ ∂ ω ∂ x μ B μ ω * ] = 1 ε ∂ ω ∂ x μ ∂ ω * ∂ x μ (A.5)

In order to analyze this relation, we first study the infinitesimal gauge, for v * = − v ,

ω = 1 + η v . (A.6)

Substituting into (A.5), one finds the condition, to first order,

{ ∂ ω ∂ x μ , B μ } = 0. (A.7)

Now, choose a local Lorentz frame for which B μ → B 0 , so that our condition becomes

{ ∂ ω ∂ x 0 , B 0 } = 0. (A.8)

For the Yang-Millls fields, we may represent

∂ ω ∂ x 0 = i a 0 + i ∑ i = 1 3 a i τ i B 0 = i b 0 + i ∑ i = 1 3 b i τ i (A.9)

where a 0 , a i , b 0 , b i are real numbers , τ i Pauli matrices. To satisfy (A.8), we must have a 0 = b 0 = 0 . What remains is the condition

∑ i n a i b i = 0 , (A.10)

closely analogous to what was obtained in (2.15) for the quaternionic theory.

We take the quaternionic Hilbert space scalar product to satisfy [9] [10]

( a f , g ) = a ( f , g ) ( f , a g ) = ( f , g ) a * (A.11)

Then, to pass to Dirac wave function representation, we use the spectral representation of the x μ operator

∫ d E ( x ) = ∫ | x 〉 〈 x | d 4 x = I , (A.12)

where the integration is in the same sense as in [17] . Then, for (conjugate of the usual form)

f ( x ) = 〈 f | x 〉 , (A.13)

we have,

( f , g ) = ∫ 〈 f | x 〉 〈 x | g 〉 d 4 x = ∫ f ( x ) g ( x ) * d 4 x , (A.14)

With our convention (A.11), we have 〈 a f | x 〉 = a 〈 f | x 〉 so that

∫ 〈 a f | x 〉 〈 x | g 〉 * d 4 x = ∫ a 〈 f | x 〉 〈 x | g 〉 * d 4 x = a ( f , g ) . (A.15)

For the left gauge ω ( x ) , it then follows that

〈 ω f | x 〉 = ω ( x ) 〈 f | x 〉 = ω ( x ) f ( x ) , (A.16)

and

( ω f , g ) = ∫ ω ( x ) f ( x ) g ( x ) * d 4 x , (A.17)

the left gauge, as we have used in the text.

Using the choice of linearity ( f a , g ) = a * ( f , g ) , with 〈 x | f 〉 = f ( x ) , we would have 〈 x | f ω 〉 = 〈 x | f 〉 ω ( x ) = f ( x ) ω ( x ) , the alternative right gauge.