# Restriction to Hamiltonians quadratic in momentum variables

The paleoconservative 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 $$\hat{p}_i | \hat{p}_i \rangle$$.

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 paleoconservative 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, momentum is also quadratic in momentum variables

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 variables. But if we use the field variables, 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 $H = \int \pi(x) \dot{\varphi}(x) d^3x - L = \frac12 \int \pi(x)^2 + \delta^{ij}\frac{\partial \varphi}{\partial x^i}\frac{\partial \varphi}{\partial x^j} + m^2\varphi^2 d^3x.$

Thus, the energy of the field is in this theory also a quadratic function of the momentum variables. Note also the simplicity of this result - we do not have to do anything non-trivial, the straightforward approach works nicely.

## 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. 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 not 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.

## 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: