Sky Nelson-Isaacs’ paper “Spacetime Paths as a Whole” addresses a fundamental challenge within the standard theory of quantum mechanics, and, to some extent, quantum field theory.

The problem to be addressed is that of time. We know that space and time should be treated essentially the same in physics, yet there is ample evidence that our models of time need revision: the Fourier domain is used ubiquitously in solving physics problems in space, but its application to time is hampered by conceptual difficulties...we cannot form a consistent “time operator”...the path integral says particles “travel” along all paths in space but along a unique path in time...Gilbert Lewis pointed out that light leaving a star many light years away will not evolve at all during its travel, as if the beginning and endpoints are the same.

For all these reasons, Nelson-Isaacs has sought in this paper to identify the inconsistency and provide a suggested solution based on the well-known engineering physics of signal processing and the Fourier transform. The approach is appealing because very little is introduced as new, but existing inconsistencies are cleared up in a novel way. An unexpected result of this reformulation is the framing of quantum mechanics in the structure of a hologram and a reasonable sense in which the world is like a computation.

Improving our model of time

A wave function is an abstract mathematical tool which describes probability in space, a map over space which changes in time. Ultimately it is a dissatisfying concept because it treats time and space differently: it is a map over all possible paths in space but only over one possible path in time, from past to future along the “arrow of time”.

The key innovation presented in this paper is to consider what it would mean to have a wave function over space and over time, in other words one which maps out all of time and does not change in time. There is difficulty in conceiving of this because as humans we think of everything as changing in time.

A “map” over space and time, when seen from the outside, is like viewing a whole path at once. It cannot undergo change, since “change” itself implies some sort of meta-time: in order for a map in time to change in time we would need an additional kind of time for the change to happen with respect to. Rather than go this route, we find instead that the wavefunction encodes the entire history (actually, all possible histories) of each object into the frequency or spectral representation. The experience of a particular moment in that history is extracted from it in the same way a particular 3-dimensional perspective is extracted from viewing a hologram at a certain angle.

One cannot define a specific “now” or “privileged present” in spacetime (or “regular space”). We say that the wave function cannot be “time-sliced.” Rather, reality is made of the possible paths for things, paths as a whole, defined from beginning to end, all at once.

This reasoning leads us to the main mathematical contribution of the paper, a distinction between two sets of symbols—parameters and coordinate intervals—for describing 4 dimensional space and time. As we will see, this is analogous to the mathematics of a hologram, in which physical behavior is encoded in a 4 dimensional frequency domain.

The frequency domain

The 4 dimensional frequency domain is an important innovation, and can be understood with practical examples. Consider the conversion of a sound into the frequency domain, like with an mp3 music file on a computer. Through application of a mathematical process known as the Fourier transform, one removes the explicit time dependence of a sound signal. Therefore, a sound file represented in the frequency domain cannot be time-sliced, for it is a map over frequencies, not over time. A given location in that file corresponds to a given frequency but has no association with a specific time. A “slice” does not give a recognizable description of any part of the original sound. This is analogous to saying there is no “now” in the regular 4 dimensional space. However, the time information is not lost. It is still present in the data, encoded into the phase profile of the signal, and it can be extracted through an inverse Fourier transform.

The picture of reality being painted here is focused on information: an abstract 4 dimensional map in space and time, with a mirroring map in 4 dimensional frequency space. In the simplest sense of the term, the world is like a holographic simulation: these two “spaces” provide an information-based description of the possible dynamical behavior or movements of things. Our movement through time is like changing our perspective on a hologram and watching the item captured by the hologram move as we shift perspectives on a hologram. The object persists, but evolves.

The “hologram” encodes possibilities, but is not itself physically real. Actual experimental results are only obtained through subjective interactions between observers. This is like what we already know from quantum mechanics, with the additional property that time is now part of this subjectivity.

Two different kinds of space and time

To picture the distinction between “parameters” and “coordinates”, it can be useful to think more about an image on a holographic film. A hologram is made of interference patterns capturing the phase from 2 dimensional frequency space onto film in regular 2 dimensional space. What makes a hologram interesting is that these 2 dimensional interference patterns generate an image with apparent 3 dimensional coordinates. When one’s vantage point on a hologram changes, the image appears to move in 3D relative to the film, i.e., the coordinates of the image change, but the interference pattern on the film (described with parameters) does not. In a hologram, and also in the theory presented here, measurable events are described by coordinates which evolve according to constraints encoded into the interference pattern described by parameters.

The interference patterns in the 4 dimensional frequency space wavefunction therefore encode information about the spatial coordinate intervals of an event in regular 4 dimensional spacetime. But an object changes due to discrete interactions only by coordinate intervals which leave the map itself (the “static block universe”) unchanged. These are “spacetime paths as a whole.” Thus, any measurable notion of time must not be considered continuous, but rather advance in discrete steps at each moment of interaction. This is a fundamental difference from our usual models in which time is experienced smoothly and objectively, from past to present to future. This is a radical rethinking of what we consider “real,” for what can be considered “real” during the in-between moments when time is not even well-defined?

Photographic image processing and holography again provides a useful illustration of the main point. It is well known within the photographic industry that the frequency domain representation of a photographic image contains a complex spectrum that has integrity as a whole and cannot be subdivided. In other words, cutting the photograph physically in half or placing a filter over the outer regions of the spectrum alters the entire complex spectrum.

The frequency domain describes relationships across images as a whole, in the same way that it describes entire histories as a whole.

Isolating a subdivision of an image in frequency space doesn’t work: it introduces errors or artifacts into the frequency domain representation, thereby altering the original image as a whole. You can see this holistic behavior in the figures. The first figure contains all of the frequencies in frequency space. In the second figure is a mask which cuts out the high frequency data, and in the third figure is the resulting image. The key idea is that this doesn’t cut the photo of the city in half, but changes its resolution (and brightness) everywhere in space.

This is an important fact in the consideration of 4 dimensional wave functions in quantum mechanics. The trajectories or paths of these signals are determined by a given 4 dimensional frequency space distribution, just as in the photographic case. Just as locally altering the complex spectrum of a photographic image in 2 dimensional frequency space generates artifacts or changes to the entire image in 2 dimensional regular space, a local modification of the 4 dimensional frequency space representation of a physical object will affect the spatial and temporal characteristics of an entire chain of events in spacetime.

To return to the definition of these two different notions of space and time, the “coordinates” constitute the usual notion of time: they are measurable and evolving. These are the events of reality, or the apparent 3D images that you see in a hologram. The “parameters” are not measurable, but still fundamentally important. They are the hidden interference information, the patterns on a holographic film that encode what you see in the film. They do not change or evolve in time, for they encode time itself.

No continuous time

Thus, only the endpoints of a path, corresponding to measurable coordinates, are describable in physical terms. Let’s consider what frequency space looks like for moving particle. It will have a given representation in the frequency domain, but containing no reference to regular space or time other than the overall interval of travel.

The frequency space expression for the particle contains no information describing continuous movement of the particle, only its overall interval of travel.

Further, the Fourier transform is unitary, meaning information is preserved during the transform between the spaces. Thus, the lack of existence of frequency space information about “continuous movement” of the particle means that such information cannot exist in regular space either. All that exists is the information about the overall interval of travel. Therefore, the interactions which begin or end a path of travel have a physical description, but the intermediate path of travel does not.

This is what Gilbert Lewis was referring to with the phrase “virtual contact” in his 1926 articles on starlight. Light leaving the Sun may take eight minutes to get to Earth, but because it is travelling at light speed, space and time do not actually exist. The notion of travel doesn’t make sense, and the beginning and end of the trip are like a single event. This is equally true of light from a galaxy that is millions of light years away. This disrupts our notion of continuous time and the present moment, and introduces timelessness and spacelessness.

The takeaway from this approach is that paths of travel are defined “as a whole” and physical descriptions can be formulated without time-slicing, without a “Now”. The motion of a system through spacetime is composed of discrete leaps between interactions rather than continuous evolution in time. A general model of reality emerges that is holistic, in other words it has integrity as a whole and cannot be subdivided. This stands in direct contrast to the broadly accepted notion of reductionism in scientific models.