On the vacuum energy in Bohmian mechanics

We consider a universe consisting of a finite number of electrons on Bohmian trajectories. We derive a quasi-vacuum solution for the Schrödinger equation of the electron-system and establish a corresponding invariant vacuum energy density Λ. The result sheds light on some fundamental issues regarding the vacuum and the cosmological constant.


Introduction
The theory of quantum fields in curved space-time is in the absence of a full-fledged theory of quantum gravity only available in a semi-classical formulation. It is governed by the Einstein equations . and R denote as usual the Ricci-tensor and Ricci-scalar, respectively. There is a significant discrepancy between the value of , L which is experimentally determined from the measured expansion of the Universe and from data of the cosmic-microwave background, and the one resulting from theoretical calculations in quantum field theory. Whereas the experimental data suggest 1, 7 10 121 L » -· Planck Units [1], the theoretical value, by whatever method it is calculated, is much larger 1 . Possible explanations for the discrepancy are that either not all the fields play a role or that the contributions cancel each other.
Because of the symmetry of the vacuum, the expectation term T 0 0 á ñ mn | | in Minkowski space must have the form with a constant 0   Î [3]. This implies Lorentz-invariance of T 0 0 á ñ mn | | 3 and energy-momentum The generalization to arbitrary metric fields T g 0 0 0  á ¢ ¢ ¢ñ = mn mn | | · still locally preserves energy 4 , because of the choice of an affine connection, but it is no longer clear, whether T 0 0 á ¢ ¢ ¢ñ mn | | still is the energy expectation of a vacuum state, since for general metric fields g mn the state 0¢ñ | does no longer necessarily represent a state with no particles, as the Davis-Unruh effect demonstrates [4,5]. There arises the question what the vacuum really signifies [3]. Einstein introduced the constant 0 L as a bare cosmological energy, without any radiation or matter fields present. Adding quantum field vacuum-energy would then amount to the with a suitable constant . h As mentioned above, there are two immediate strategies 5 to explain the measured contribution E vac h á ñto . L Either one shows that some contributions cancel each other, or that some fields do not play a role. Supersymmetry would do the job along the first strategy, but it is clearly broken in nature. We take a path along the second one. In order to approach the question of E vac h L = á ñwe can look at the ontology of quantum physics. There is one interpretation, which allows to select some fields. Our choice for the 'beables' is persistent fermionic particles on Bohmian trajectories [6]. We make this choice for different reasons. First of all, because it is a simple ontology, which clearly decides that the primary reality is matter and by taking the fermion sectors of the standard model we cover the bulk of it. Second, because it clarifies the status of space, which is a property, a relation between pieces of matter. Finally, by taking the particle-ontology we omit the fallacies of the corresponding field-ontology [7]. We will in this paper rigorously develop the model for the electron sector. Very few rigorous results of the Bohmian evolution of quantum fields are known due to the complicated nature of the equations [6]. We can, however, make qualitative statements subsuming e.g. their interaction with the electron sector effectively in a time dependent 'external' potential. We will thus work in a universe with a number of N electrons, moving on Bohmian trajectories. We consider electromagnetic and gravitational forces acting on them in such a way, that the guiding equation evolves in a, yet to be defined, quasi-vacuum and then derive L in this model, by bridging in a suitable manner from Minkowski space to general manifolds and showing that the resulting energy tensor is of the form (3).

The electron gas
2.1. The setting We will use the Bohmian picture, as done in [6], and think of the electron gas as a large number N  Î of electrons, which move along trajectories defined by a spinor guiding-field t , for the velocity of the kth particle and with guiding equation for the field t and b are 4 4 ´-matrices, which can be expressed in terms of the Dirac The Dirac matrices satisfy the known commutator-relations 2 .
k l l k kl The space-like components of H EM k constitute the magnetic interaction induced by the charges, which move by where m e is the electron mass. In order to ensure that all the components of the Hamiltonian H N are well defined, we must as usual assume an ultraviolet cut-off, implying for the momenta p and an infrared cut-off, implying for position x R.  | | Finally, we can model the effects of the other sectors of the standard model by use of an effective potential ) We will begin by making the assumption that V t x , 0 = ( ) and work with product states t .
To justify this we will construct an approximate vacuum-type solution t (5), (6) such that no mixing occurs.

Approximate solution
It is a useful strategy [6] to try to simplify the system of equations (5) To find a candidate-solution of (15), we first search for 0 W such that Since the Dirac operators H k 0 commute pairwise, we find 0 W and E , 0 as explained in [6], by filling successively the N lowest negative eigenstates , and since there are two states (spin up and down) for each energy level, we will have a minimum for N 2 , In out context the vacuum is not a situation with no particles but a situation, where a system of N free particles does have minimum energy by filling all the negative energy-states available and the particles are equally distributed within the region B .  (15). 6 We work in the Lorentz-gauge.
Let us calculate (17) for each interaction term in (7) individually. Due to the normalization constants and the definition of the inner product on , and because of the law of large numbers, (21) is (approximately) a constant E C for N large enough. The potential of a uniformly distributed charge e n 2 w | | over a sphere of radius R at distance The terms in the number E B k are depending on n and and hence there is no straightforward application of the law of large numbers at the level of an individual component k. It is, however, possible to control E ,

Now we can apply the law of large number to get
We have shown so far, that it is with an additional assumption indeed possible to look at equation (15) instead of (6) and to find a solution t W (18), which defines a quasi-vacuum. Each particle moves, as if it were alone, and the particles remain equally distributed over B . R Because of this symmetry we still expect that T . 0  h á ñ = mn mn · We further see that the probabilities t t áW W ñ | are Lorentz-invariant, whereas the single energy terms E i were derived in a preferred frame. At the level of predictions the theory is relativistic, but at the more fundamental level it is not. Hence the expression can't yet (locally) serve as the energy tensor T á ñ mn in (2)/(3). In addition we remember that, different to the vacuum-energy of the matter-field E , 0 we have not yet taken into account the energy contribution from the radiation-vacuum, which amounts to E h 2 n = n per mode .
n Having this in mind, we will seek to bridge from quantum mechanics to general relativity in the second part of the paper and find an invariant . L

Energy terms and coupling-constants
Based on the results in section 2, we calculate the acceleration of a test-electron, which is positioned at the spatial-edge x R = | | of the model-universe. In [9] a universal observer is introduced, who measures the duration until an event has happened by means of a thermal clock. In order to have consistent durationintervals, accelerated observers have to synchronize (gauge) their clocks and in 10 [ ]it is shown that the synchronization of a thermal clock in the local rest-frame of an observer with acceleration , k and a single-mode vacuum light clock 8 with energy E h 2 n = n leads independently of n to the relation In (31) k B and h denote, as usual, the Boltzmann and Planck-constants. The temperature T k is hence which is the Unruh-Davies temperature. Relations (31) and (32) will now serve as the link between quantum mechanics and general relativity.
Assume that we work in a local Minkowski space, namely the local rest frame of a test-electron at distance In paragraph 2 we calculated the energies E E , ( ) 8 We mean in this context rather the core of a clock, since we do not need devices to indicate time. 9 In order to later on define a four metric by (33), we multiply the accelerations for dimensional reasons by the Lorentz-scalar .
Making use of (32) and simplifying notation E k Q e E , Analogously we get for g R the following chain of expressions Arranged differently and with the Planck length l G c By again making use of (32) we finally get in analogy to (37) has the necessary features of an energy tensor (3). It is invariant and the divergence is zero. The energy terms (and coupling-constants) were explicitly constructed by considering the Unruh effect.

Cosmological constant
The scalar factor in equation (42)

Discussion
We made the assumption in paragraph 2.1 that the effective potential V t x , 0. = ( ) If this is not the case, then it is show in [6] that what happens is, that some components of the vacuum state t W develop into excited states above the ground state .
In the Einstein equations the energy contribution by q will enter the ordinary, matter-induced energy term T .
á ñ mn So we can expect over time that the number of particles following a vacuum trajectory t w is going to decrease, N t N, < ( ) and hence t L = L( ) is timedependent and decreasing. We developed the equations in the intuition of the Bohmian ontology, where permanent particles move in space and distances are relational properties [13]. If during this process the spatial distances increase and hence , t 0 S > S then this fact is another source for a decreasing cosmological factor (30).
The explicit contributions of the evolution of other fermionic sectors to the vacuum energy L are beyond mathematical reach at the moment, simply because the system of equation (5) is very complicated. Due to the structure of the underlying forces (e.g. asymptotic freedom), we would however conjecture that there might be similar vacuum contributions by other leptons, behaving similarly to electrons, but that any vacuum states for quarks very quickly develop into ordinary matter contributions above a ground state, which is indeed in confirmation with recent results [14]. Observed data today are interpreted as the result of an accelerated, expanding space-time, making a cosmological term necessary. The model we have chosen helps to show why there is a cosmological factor L at the first place and what its sources might be, namely some of the fermion sectors in quantum field theory. In addition we saw that this cosmological factor, being a percentage of the number of excited particles N t N N t , -( ) ( () ) is time-dependent and decreasing, which can help to explain the small value of L today. In fact, in the theory of an expanding universe the constancy of L has to be assumed a priori and can only be explained in our model universe by the introduction of a bare cosmological constant 0 L (4), which has no relation to any known fields.