Finsler geometry and quantum field theory

Howard E.Brandt

U.S. Army Research Laboratory, Adelphi, Maryland, U.S.

hbrandt@arl.army.mil

Abstract. Finsler geometry motivates a generalization of the Riemannian structure of space-time to include dependence of the space-time metric and associated invariant tensor fields on the four-velocity coordinates as well as the space-time coordinates of the observer. It is then useful to consider the tangent bundle of space-time with space-time in the base manifold and four-velocity space in the fiber. A physical basis for the differential geometric structure of the space-time tangent bundle is provided by the universal upper limit on proper acceleration relative to the vacuum. It is then natural to consider a quantum field having a vanishing eigenvalue when acted on by the Laplace-Beltrami operator of the space-time tangent bundle. On this basis a quantum field theory can be constructed having a built-in intrinsic regularization at the Planck scale, and finite vacuum energy density.

Finslerian fields

A physical Finslerian field F(x.v) is one that depends not only on the observer's space-time coordinates,

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ x ≡ {x^m}= {x⁰, x¹, x², x³},љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (1)

but also on the observer's four-velocity coordinates,

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ v ≡ {v^m}= {dx^m/ds} = {v⁰, v¹, v², v³},љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (2)

where ds is the infinitesimal interval along the world line of the observer [1]-[3]. The four-velocity coordinates play here the role of the tangent space coordinates of Finsler geometry. It can be argued that the space-time-metric field g_mn itself must in general depend not only on where it is observed in space-time, but also on the four-velocity of the 'observer', namely it is a Finslerian field [1]-[3]:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ g_mn = g_mn (x,v).љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (3)

(The reader may prefer to replace the word 'observer' by 'measuring device', 'object acted upon by the field', or 'some other field interacting locally with the field'.) The space-time metric in a canonical Finsler space-time is not only Finslerian, but also satisfies special homogeneity conditions involving the dependence of the metric on the tangent space coordinates, v. In particular, one has [4], [5]

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ ds = L(x, dx),љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (4)

where L is the fundamental Finsler function, and

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ L(x, adx) = aL(x, dx),љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (5)

from which it follows that

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (6)

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ L²(x,v) = g_mnv^mvⁿ,љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (7)

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (8)

and

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (9)

In considering Finslerian space-time and associated embedded Finslerian fields, it is useful to consider the tangent bundle of space-time with space-time in the base manifold and four-velocity space in the fiber. Using the homogeneity relations, Eqs. (4)-(9), then the connection and Riemann curvature scalar of the space-time tangent bundle [6] can be significantly reduced for the case of a Finsler-space-time base manifold [4], [7]. However, the special homogeneity requirements may not hold physically in general, but may only hold in certain special space-time models.

A physical basis for the differential geometric structure of the space-time tangent bundle is provided by the universal upper limit a₀ on proper acceleration a relative to the vacuum [8]-[11]. If the proper acceleration were sufficiently large, then, because of vacuum radiation in an accelerated frame (in which particles are produced with average energy proportional to the proper acceleration), particles would be produced with mass such that their Schwarzschild radius exceeds their extent (Compton wave-length). Copious production of black-hole anti-black-hole pairs would ensue, accompanied by breakdown of the classical space-time structure, and the very concept of acceleration would loose any meaning because of the resulting complex topology. Explicitly, the maximal proper acceleration, a₀, is given by [8]

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (10)

where a is a number of order unity, c is the speed of light, љis Planck's constant divided by 2p, and G is the universal gravitational constant. This is the maximum possible proper acceleration relative to the vacuum and is taken to be universal. Hence for any proper acceleration a, one requires

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (11)

But, according to the differential geometry of space-time, the proper acceleration, a, along a world line in curved space-time is given by

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (12)

where the four-velocity v^m is given by

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (13)

and љdenotes the covariant derivative of the four-velocity with respect to the interval along the world line, namely,

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (14)

in which љis the space-time affine connection, and

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ ds² ≡љ g_mn dx^m dxⁿљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (15)

is the line element of space-time. Then substituting Eqs.(12) and (14) in

Eq.(11), one obtains

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљ (16)

Next substituting Eqs. (13) and (15) in Eq.(16), one obtains [10]

љљљљљљљљљљљљљљљљљ љљљљљљљљљљљ (17)

where

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (18)

is the minimum radius of curvature of world lines. Equation (17) defines the eight-dimensional quadratic form ds², which is non-negative along the world line. The inequality, Eq. (17), simply expresses the fact that the proper acceleration can never exceed the maximal proper acceleration. By analogy with the construction of the space-time line element of general relativity from the limiting speed of light, it is natural to take ds² to be the line element in the tangent bundle of space-time, in which the space-time coordinates x^m are the coordinates in the space-time base manifold, and the four-velocity coordinates r₀v^m (modulo a factor of r₀) are the tangent space coordinates.

The bundle line element, Eq.(17), can be rewritten as follows [10], [1]:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ ds² ≡љ G_MNdx^Mdx^N,љљљљљљљљ {M,N = 0,2,...,7},љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (19)

where the bundle coordinates are

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ {x^M} ≡љ {x^m, r₀v^m} ,љљљљљ {M = 0,2,...,7; m = 0, 1, 2, 3},љљљљљљљљљљљљљљљљљљљљљљљљљљљ (20)

and the metric of the tangent bundle of space-time is

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (21)

in which

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (22)

The bundle metric G_MN, given by Eq.(21), has a structure similar to that of an eight-dimensional Kaluza-Klein gauge theory in which the higher dimensions are in four-velocity space, and љis the gauge potential. Eqs.( 19)-(22) served as the starting point for investigating possible implications of a limiting proper acceleration for the differential geometric structure of the tangent bundle of space-time [1]-[4], [6], [8]-[24]. Possible forms for the bundle connection, curvature, and action were explored, including those based on Riemann and Finsler space-times, and also Kähler and complex space-time tangent bundles. Among the many differential geometric invariants of the space-time tangent bundle, important for the present discussions is the Laplace-Beltrami operator:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (23)

This is the invariant generalization of the wave operator, or d'Alembertian, of field theory. A simple invariant field equation for a Finslerian scalar field f(x,v) is then given by [20]

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ Lf (x,v) = 0.љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (24)

Finslerian scalar quantum fields

When the space-time is Minkowskian, the ordinary inhomogeneous Lorentz group (or Poincaré group) is a subgroup of the invariance group of the space-time tangent bundle [12], [20]. It is of interest to examine quantum field solutions to Eq.(24) for this simple case, in order to establish connections between the Finslerian framework and canonical relativistic quantum field theory. To this end, consider a simple case in which the space-time is completely flat. In particular, take the space-time metric to be Minkowskian. For this case, it was argued in earlier work that the scalar field satisfying Eq.(24) is given by [20], [25], [26]:

љљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљ (25)

where p denotes the four-momentum p^m = {p⁰, p¹, p², p³} of a particle excitation of the scalar field, a^Љ(p) and a(p) are the particle creation and annihilation operators satisfying the commutation relations

љљљљљљљљљљљљљљљљљљљљљљљ [a(p), a^Љ(p′)] = d ³(p - p′),љљљ [a(p), a(p′)] = 0,љљ [a^Љ(p), a^Љ(p′)] = 0,љљљљљљљљљљљљљљљљљљ (26)

d ³(z) is the three-dimensional Dirac delta function, and q₁(z) is the Heaviside function [28],

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (27)

Also in Eq.(25), N is a normalization factor such that the field operator acting on the vacuum state, namely , is the state of the field corresponding to a single-particle excitation appearing at some point (x,v) in the space-time tangent bundle, namely [25], [26],

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (28)

where V denotes the volume of space, m is the mass of the scalar particle excitation of the quantum field, and K₁(z) is the modified Bessel function of the third kind of order one.

It can be shown that both the positive and negative frequency terms in Eq.(25) are proportional to [20], [9]:

љљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљ (29)

where dx/dt is the ordinary velocity of the observer, and

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (30)

As can be seen from Eq.(29), there occurs an intrinsic Planck-scale regularization of the quantum field, with an exponential energy cutoff beyond the Planck energy.

It is important to note that for an observer with ordinary velocity

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (31)

in Minkowski space-time, one has

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (32)

and therefore

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (33)

Thus the ordinary velocity љcorresponds to both future and past directed four velocity. This is essential in order that both the positive-frequency particle modes and the negative-frequency antiparticle modes in Eq.(25) and in the following have non-vanishing support. One is reminded of the Stückelberg-Feynman idea that antiparticles can be represented as particles with proper time reversed relative to true time [21], [27].

Hamiltonian of scalar quantum field

On the basis of microcausality, it was argued in earlier work [21] that it is logical to define the generalized adjoint (f (x,v))^Љ of the quantum field f (x,v) by

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (34)

in which the ordinary adjoint is taken, but also the sign of the four-velocity v is changed. This is connected, through charge-conjugation, with the fact that the positive-frequency particle component of the field in Eq.(25) has non-vanishing support for positive v⁰ (or equivalently, positive pv), and the negative-frequency anti-particle component has non-vanishing support for negative v⁰ (equivalently, negative pv) [21]. In order that the Hamiltonian be Hermitian (in terms of the generalized adjoint), that it reduce to the canonical form in Fock space, and that the observer's four-velocity lie on the four-velocity shell (v² = 1), it is natural to define the Hamiltonian for the scalar quantum field as follows [25], [26]:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (35)

in which the bundle energy density T₀₀(x,v) at point (x,v) in the bundle is integrated over the entire space-time tangent bundle with large spatial volume V, d (z) is the one-dimensional Dirac delta function, and the bundle energy density is given by

љљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљ (36)

To see that the Hamiltonian is Hermitian, one first notes that

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (37)

Next, taking the generalized adjoint of T₀₀(x,v) and using Eq.(34), one obtains

љљљљљљљљљљљљљљљ љљљљљљљљљ (38)

Equivalently, using Eq.(25), Eq.(38) becomes

љљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљ (39)

or comparing Eqs.(39) and (36), one obtains

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ T₀₀(x,v)^Љ = T₀₀(x, -v).љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (40)

Next substituting Eq.(40) in Eq.(37), one has

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (41)

and replacing the dummy variable of integration v by -v, Eq.(41) becomes

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (42)

Finally, substituting Eq.(35) in Eq.(42), one concludes that

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ H^Љ = H.љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (43)

Thus H^Љ is in fact Hermitian.

Next, using (25), (27) and (36) in (35), integrating over space, and substituting the integral [25], [29]

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (44)

one obtains the canonical expression for the Hamiltonian operator for a scalar quantum field in Fock space [25]:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (45)

in whichљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (46)

For consistency, it is also well to verify that the Heisenberg equation of motion for the field f is satisfied, namely [30],

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (47)

To see that (47) holds, one uses (45) and (25) to evaluate the commutator [H,f]. Thus one has

љљљљљљљљљљљљљљљ љљљљљљљљљ (48)

or equivalently,

љљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљ (49)

Then substituting Eqs.(26) in Eq.(49), one obtains

љљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљ (50)

Finally, substituting Eq.(25) in Eq.(50), one obtains the Heisenberg equation of motion for the field, Eq.(47).

Vacuum energy density

Using Eqs.(35) and (36), and denoting the vacuum state by , it follows that the vacuum energy in the bundle, for spatial volume V, is given by [25]:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (51)

which is the canonical divergent result [30]. However, a particular observer has four-velocity v, and his world line at any time is confined to the neighborhood of v in the fiber (See Eq.(33)). The vacuum energy which he observes is therefore given by [25]

љљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљ (52)

in which D³v is defined by

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (53)

where d v is the spread in the spatial components of the four-velocity of the observer, and because of the Dirac delta function in Eq.(52),

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (54)

One notes that in Eq.(52), one can use the well known identity:

љљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљ (55)

Equation (53), defining the spread, is true because the expression following D³v in Eq.(52) turns out to be proportional to (1/v⁰) = (1+|v|²)^-1/2, with no other dependence on the spatial component v of four-velocity (See Eqs.(54) and (62)-(67)).

Next, denoting the three respective contributions to Eq.(52) of the three terms of Eq.(36) by , and , respectively, then

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (56)

Then using Eqs.(36), (52), and (56), one has

(57)

љљљљљљљ .љ

and substituting Eqs.(25) and (55) in Eq.(57), one obtains

(58)

Here q₁(+r₀pv/ÿ) has been replaced by q₁(+v⁰), since it can be shown that pv > mc when v⁰ = gљ > 0, and pv < mc when v⁰ = -g < 0 [20], [21]. Then, according to Eqs.(27), (30) and (54), one has

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ q₁(v⁰)q₁(-v⁰) = 0,љљљ [q₁(v⁰)]² = q₁(v⁰).љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (59)

Next using Eq.(59) in Eq.(58), one obtains

љљљљљљљљљ љљљљљљљљ(60)

One also has

љљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљ (61)

Therefore using Eqs.(61) in Eq.(60), one obtains

љљљљљљљљљљљљљљљљљљљљљљљљ (62)

or equivalently,

љљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљ (63)

Analogously, corresponding to the contributions to Eq.(56) of the two remaining terms of Eq.(38), containing derivatives with respect to space and four-velocity, respectively, one obtains

љљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљ (64)

and

љљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљ (65)

Then substituting Eqs.(63)-(65) in Eq.(56), noting that

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ p⁰² = |p|² + m²c²,љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (66)

evaluating the following integral [26], [29],

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (67)

where K₂(z) is the modified Bessel function of the third kind of order 2, substituting Eq.(28), and dividing by the spatial volume V, one finally obtains the vacuum energy density seen by an observer at (x,v) in the space-time tangent bundle:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (68)

Thus any one observer sees a finite vacuum energy density given by Eq.(68). One notes that for sufficiently small spread in the spatial part of the observer's four-velocity (Eq.(53)), the observed vacuum energy density may be near vanishing. This may be consistent with a near vanishing cosmological constant.

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Howard E.Brandt, Researcher-scientist (Army Research Laboratory, Adelphi, USA); he is a theoretical physicist with primary scientific interests in quantum field theory, general relativity, differential geometry, and quantum information processing. The present paper is an expanded version of an invited paper, "Finsler Geometry and Quantum Field theory", presented at the Fourth World Congress of Nonlinear Analysts in the session: Applications of Finsler Differential Geometry (in Engineering, Physics, and Biology) held July 5, 2004, in Orlando, Florida, U.S.