# Restriction to Hamiltonians quadratic in momentum variables

The minimal realistic interpretation makes, in comparison with the minimal interpretation, an important assumption about the Hamilton operator: It should be quadratic in the momentum variables: $\hat{H} = \langle \hat{p}_i | \hat{p}_i \rangle + V(\hat{q}^i).$ That means, while the function $$V(q)$$ can be an arbitrary smooth function on the configuration space, the dependence on the momentum variables is much more restricted - it should be a quadratic function $$\langle\hat{p}_i | \hat{p}_i \rangle$$.

The classical example for such a Hamilton operator is that for several non-relativistic point particles with masses $$m_i$$. $\hat{H} = \sum_i \frac{\hat{p}_i^2}{2m_i} + V(\hat{q}^i).$

This assumption the minimal realistic interpretation shares with all other realist interpretations, in particular with de Broglie-Bohm theory, Nelsonian stochastics, and Caticha's entropic dynamics. Therefore it is a quite popular argument against all these interpretations that they are non-relativistic, unable to handle relativistic effects. But this is simply wrong.

## In relativistic field theory, energy is also quadratic in momentum variables in a field ontology

At a first look, this seems to restrict the interpretation to non-relativistic theory. Indeed, for a relativistic particle, the energy is no longer a quadratic function of the momentum variables, but defined by $$E = \sqrt{m^2c^4 + p^2}$$.

Fortunately, the situation is different in relativistic quantum field theory. In field theory, the natural configuration space is not defined by particles, but by the field configurations. But if we use the field configurations, then the energy is again quadratic in the momentum variables.

## The case of the scalar field

Let's consider the simplest case of a scalar field $$\varphi(x)$$. The Lagrangian is $L = \int \mathcal{L} d^3x = \frac12 \int \dot{\varphi}(x)^2 - \delta^{ij}\frac{\partial \varphi}{\partial x^i}\frac{\partial \varphi}{\partial x^j} - m^2\varphi^2 d^3x.$

The momentum is defined as $\pi(x) = \frac{\delta L}{\delta \dot{\varphi}} = \dot{\varphi}(x).$

For the energy we obtain $H = \int \pi(x) \dot{\varphi}(x) d^3x - L = \frac12 \int \pi(x)^2 + \delta^{ij}\frac{\partial \varphi}{\partial x^i}\frac{\partial \varphi}{\partial x^j} + m^2\varphi^2 d^3x = \frac12 \int \pi(x)^2 d^3x + V(\varphi(.)).$

Thus, the energy of the field is in this theory also a quadratic function of the momentum variables plus a functional V which depends only on the configuration variables $$\varphi(x)$$, not on their time derivatives. Note also the simplicity of this result - we do not have to do anything non-trivial, the straightforward "nonrelativistic" formulas work nicely.

A side effect of this is that all the "non-relativistic" interpretations which rely on this structure of the Hamilton operator (that means all those which use the so-called "hydrodynamic variables", in particular de Broglie-Bohm theory, Nelsonian stochastics, Caticha's entropic dynamics and what I propose here) can be applied to such relativistic field theories too.

## The lattice theory as a canonical quantum theory

Of course, this theory has all the problems connected with an infinite number of variables $$\varphi(x)$$ in quantum field theory. But using a lattice discretization on a cube with periodic boundary conditions we can approximate this theory by a canonical quantum theory with a finite number of degrees of freedom $$\varphi_n = \varphi(x_n)$$. Such a lattice approximation is sufficient, the only problem is the conceptual problem that it has no exact relativistic symmetry. But an approximate relativistic symmetry is completely sufficient for all practical purposes.

Let's note that the use of the field variables $$\varphi(x)$$, which in a lattice approximation become the field values on the lattice nodes $$\varphi_n = \varphi(x_n)$$, automatically avoids all problems with particle creation and destruction, which appear if one uses particles to define the configuration space. Lattice nodes do not have to be created or destroyed.

## The case of gauge fields and Dirac fermions

To describe all the fields of the standard model of particle physics, one has to be able to handle also more complicate relativistic fields, in particular gauge fields and Dirac fermions.

For gauge field, the simplest way to handle them is to ignore gauge symmetry. In this case, the gauge field becomes simply a vector field. The gauge degrees of freedom do not interact with anything else, thus, they will be unobservable dark matter, and adding such dark matter does not cause problems.

To obtain Dirac fermions is more complex. One possibility is the following: One starts with a scalar field, but with a degenerated vacuum state. This does not change anything relevant, thus, the method described here can be used. Then one considers only the lowest energy states in a lattice approximation. In each lattice node, there will be two lowest energy states, defined by the symmetric and the antisymmetric combination of the ground states of the two minima. Their energies will be very close to each other, while there will be a large gap to the next higher energy states. Thus, we obtain de facto a $$\mathbb{Z}_2$$-valued lattice theory.

Such a $$\mathbb{Z}_2$$-valued lattice theory, with a spatial lattice but continuous in time, appears equivalent to a variant of staggered lattice fermions, which gives, in the continuous limit, two Dirac fermions. Fortunately, Dirac fermions appear in the standard model only in pairs, thus, it appears unproblematic that this construction gives only pairs of Dirac fermions. For the details of this construction see my paper:

## Problems with relativistic gravity

For relativistic gravity also exists a condensed matter (ether) interpretation – the General Lorentz Ether Theory. This ether theory has been published in a peer-reviewed journal:

In a natural limit, it gives the Einstein equations of general relativity, and, as a consequence, the ether theory appears compatible with the empirical evidence supporting GR. It describes an ether which follows the continuity and Euler equations known from classical condensed matter theory.

But for a classical condensed matter theory one can construct atomic models so that the energy of the atoms is quadratic in the momentum variables.

## Incompatibility with relativistic metaphysics

So there is no are no physical objections left: For all experimentally established parts of modern physics there exist theories with preferred frame.

There remains the ideological part – relativistic metaphysics, which requires that relativistic symmetry cannot be simply an approximate symmetry, but has to be of fundamental character.

This position was a quite reasonable one at Einstein's time. But after Bell's theorem and the experimental confirmation of the violations of Bell's inequality it is no longer reasonable. To reject realism out of some metaphysical preference for theories with nice symmetry comes close to absurdity. I explain this in more detail in my defense of realism pages.