Blocked Temporal Evolution

Before anything else, the reader should be warned that this section contains material which is of a speculative nature, currently unsupported by either calculations or simulations. We mean here only to suggest an idea of how one of the remaining problems with the structure of the theory could be solved, thus seeding ideas for future research. No more than some intuitive reasoning will be offered here in support of the ideas presented.

One of the main remaining open problem in the basic structure of our formalism is the representation of temporal evolution. Now, one must realize that the loss of the Hilbert space formalism does not necessary imply the loss of the concept of temporal evolution, but only the loss of its usual representation within that formalism. We mean to propose here the substitution of the sharp-time temporal evolution of non-relativistic quantum mechanics by a blocked-time temporal evolution, which we will describe qualitatively on the Euclidean lattice. Of course, in order to represent temporal evolution in Euclidean space and be able to analyze it in any kind of detail, one would have to first understand in more detail than usual the relationship between the dynamics of the Euclidean and Minkowskian theories. However, here we will just propose the idea and ignore any such concerns.

On a fundamental level temporal evolution is a relationship between measurements made at different times. Since all measurements can only be done in the complete extent of $d$-dimensional regions of space-time, we will define the temporal evolution in terms of $d$-dimensional boxes. The idea is that temporal evolution will be a relationship between two consecutive $d$-dimensional regions of space-time, each holding a copy of a local quantum state. The transmission of information between them will be done through a $(d-1)$-dimensional surface, which is the interface between the two consecutive regions. We will propose the idea in the context of two identical lattices, each one contained in one of the two boxes, using the context of a stochastic simulation of the resulting system as a way to illustrate the ideas.

Figure 6.2.1: The lattice model proposed as a representation of the temporal evolution of blocked quantities.

The drawing in figure 6.2.1 may help the reader to visualize the proposed system. In this figure the sets of $8$ sites connected in circles are representations of the temporal directions of two $d$-dimensional lattices. The spacial dimensions of the lattices are omitted for simplicity of the drawing. In order to simplify the treatment of the boundary conditions we will adopt periodical boundary conditions for each lattice. Note that there is no problem involved with the use of the periodical boundary condition in the temporal direction within each box, because this internal temporal variable does not represent temporal evolution. The middle arrow represents the interface between the two lattices, connecting a $(d-1)$-dimensional spacial section of the left lattice with a corresponding spacial section of the right one.

The $(d-1)$-dimensional surface interfacing the two boxes will become an arrow of time, establishing a temporal ordering between the two boxes, in the following way: the initial box will contain a full realization of some state of particles, that is, a stochastic simulation of the corresponding statistical distribution; one then takes the field that results from this distribution, on the chosen $(d-1)$-dimensional space-like surface perpendicular to the temporal variable of that box, and copies it dynamically into a corresponding space-like surface of the final box; within the second box one builds a direct stochastic realization of the statistical distribution of another physical state, typically the vacuum state, at all sites except for this $(d-1)$-dimensional space-like surface, which will function as a dynamical boundary condition for the rest of the lattice within the second box.

If the state in the first box is, say, a one-particle state with momentum $\vec{k}$, then the second box will be subject to the effect of a $(d-1)$-dimensional surface containing a section of that first ensemble. This will affect the distribution within the second box, which will no longer be simply the vacuum. In this way the physical situation within the first box can propagate into the second box, along what we may call Monte-Carlo time, in a type of diffusion process, leading eventually to an equilibrium situation which represents the physical propagation of the state. The physical propagation is expressed as the difference between what we implemented directly in the second box (the vacuum state) and what eventually turns up within it as a consequence of the influence of the first box (in this example, possibly a one-particle state). Note that the cause-and-effect relationship is only from the first box to the second box, never the other way around, and hence we see that this scheme indeed implements an arrow of time.

Let us now discuss why would one think that such a $(d-1)$-dimensional boundary could have such a large effect, over the whole interior of the second box. The fact is that the effect of a $(d-1)$-dimensional boundary over a $d$-dimensional region of space is usually very large. One can consider, for example, the classical static case of electrical charges distributed over a two-dimensional surface: such a two-dimensional plane of surface charge will fill homogeneously the whole three-dimensional space with an electric field, while a line of charge and a point charge have an effect that decays with the distance. Contrary to what seems to be the popular belief in some quarters, the same is true in the quantum case. If a $(d-1)$-dimensional surface is covered with sources, then what is left as a propagation direction is the single remaining dimension, and in that direction there is therefore no solid-angle, that is, there is no angular increase with distance to promote the damping of the influence of the sources.

In the second box the $(d-1)$-dimensional boundary surface acts very much like an external source, except for the fact that it is dynamical and not static, that is, it is a fluctuating source with a dynamics comparable to the internal dynamics of that lattice. Hence, what will propagate from it into the second box is not a static field but a probability distribution of values for the field, which will therefore affect the distribution within the second box. It is interesting to note that this is a new kind of “fixed” boundary condition, in which the values of the field itself are not fixed, but the distribution of the values of the field is a fixed and given one. This is the only kind of fixed boundary condition that is physically realizable in the quantum theory, because one can never really fix the values of the intrinsically fluctuating fundamental field.

Although it should in principle be possible to deal with this whole scheme by analytical means in the Gaussian model, since all distributions are Gaussian, we currently do not know how to do this. The only treatment currently available would be by means of stochastic simulations, which puts the subject outside the scope of this book. But we may describe how one would go about doing this analytically. In order to determine which probability distribution should be implemented within the $(d-1)$-dimensional boundary surface of the second box, one must take the $d$-dimensional ensemble within the first box and integrate out all the variables except those within the chosen $(d-1)$-dimensional interface, thus producing an explicit representation of the distribution over this surface, which is a consequence of the distribution within the whole box. One can then use this distribution as a boundary condition for the second box.

Figure 6.2.2: The lattice model proposed as a representation of a scattering process.

It is currently difficult to guess any details about this new type of temporal evolution, but it is a distinct possibility that propagation into the second box may depend on the nature of the state in the first box, for example on whether it satisfies or fails to satisfy the on-shell condition. It is reasonable to think that only on-shell states should propagate, specially in the continuum limit. Note that on-shell waves necessarily correspond to modes with time-like momenta and hence have wave vectors pointing more directly through the $(d-1)$-dimensional interface. However, definite answers to any of the many questions one could ask about this idea will have to wait for the results of further research, probably involving some large-scale stochastic simulations of the proposed system.

If this idea turns out to work, then one can readily imagine other uses for it, for example the direct representation of scattering processes, as illustrated in figure 6.2.2. Here we have three consecutive boxes, the initial one holding an initial state of two particles, say with momenta $\vec{k}_{1}$ and $\vec{k}_{2}$. The middle box is an interaction region, where an interaction such as one finds in the $\lambda\phi^{4}$ polynomial model is turned on, and the parameters of the model are tuned so that the physical mass of its particles is equal to the physical mass of the incoming free particles. The third box contains the free vacuum and is a detection region, where one will look to find out what particles with which momenta show up. This would be a discrete realization of the scattering structure which is usually represented by asymptotic “in” and “out” fields and states in the traditional approach to the subject.

One can go further on with ideas like this, for example proposing the definition of thermal states, by starting with a state containing some set of many particles, and propagating it through a large number of consecutive interaction regions, until the set of particles finds a true thermal equilibrium distribution, that is, a distribution of particles that no longer changes when it passes through an interaction region. For practical reasons one could adopt the following alternative approach: instead of a long series of interaction regions, use only one interaction region and add to the system a feedback mechanism from the third box to the first box, that modifies the initial distribution of particles so that it converges to the final distribution of particles which is detected in the third box. In any case, one can see that there is plenty of field for exploration and further research in this subject.