Interview with Sky Nelson-Isaacs, “Spacetime Paths as a Whole”, on Clubhouse
June 5, 2021
Sky Nelson-Isaacs 0:00
My work has been for many years trying to understand some basic questions in space and time, and the foundations of quantum mechanics, and I went into education for a while, I’ve spent many years as a high school teacher. I’ve also worked in industry and software. And currently, the paper “Spacetime Paths as a Whole” that I was speaking about was published in quantum reports in December, and as a result of funded research that was done for the previous two years. So happy to talk about it today. Thank you for having me.
Paul Borrill 0:33
Thank you for being here, Sky, it’s a pleasure to have you. So let’s get started on the paper Spacetime Paths as a Whole. So my first question here is can you lay out the basic model, what motivates it, and what is it intended to accomplish?
Sky Nelson-Isaacs 0:50
I was inspired really early in my education in college by learning about holograms, learning about Fourier transforms. I was really interested in fractals and all these things held sort of a similar mystique to me. I got to build a hologram in college, so I really got a first hand experience of what that is and what’s amazing about a hologram is that it encodes the information in a distributed fashion, so you’re taking a photograph of like, say, I took a photograph of a rubber duck, you know, we had this big table, and add a rubber duck in the middle of the table, turned off all the lights and shone a laser on it, and captured the light bouncing off of the rubber duck with a piece of film.
And in that film, the duck looks realistic and looks three dimensional. What’s amazing about it i s if you cut the hologram in half, you still see the whole duck, because the duck is not actually fixed to one part of the film it’s actually distributed across the film. So depending on your perspective on the film, you see the duck in different places and different angles. So there’s a disconnection between the space of the film itself and the image you see in it. And there’s a holism that happens, a one to all relationship that was evident in the Fourier transform that describes the hologram, and in the physical hologram itself.
And to me that was a fundamental thing that wasn’t fully explored. You can see this in this example, too and anybody’s familiar with the ability to break down an image into pixels, so your screen is made of pixels that are all different colors, and they’re side by side so they create an image together. But you can also decompose any image in terms of frequencies, in terms of patterns that look kind of like zebra stripes going in all sorts of different directions. And this is the Fourier transform. And what’s amazing about that is that you can, with the same amount of data but using these really different fundamental building blocks, you can create the same imagery, and what’s different about a point particle or like a pixel and a waveform is that a waveform is distributed across all of the space and infinity. So the question of how do you decompose things into these non local patterns, I think is really foundational.
And, turning to quantum mechanics, there’s some basic questions in quantum mechanics that I’ve had for a long time, specifically around time. When you go from the Schrodinger equation in its simplest form called the time independent equation, Time is sort of very simple, there’s a simple way of looking at that and it’s very classical. And when you go to the time dependent Schrodinger equation suddenly gets very complicated, you have to add these recursive cycles of computation.
So it seemed to me early on that there’s more than just trying to make special relativity compatible with quantum mechanics, there’s actually some, some other aspects of time that we don’t fully understand. And with my interest in optics and images and decomposing images, I thought that there might be some way to connect that field with the field of quantum wavefunction propagations. So my paper is not really a reformulation of quantum mechanics. It’s a reformulation of just wave function propagation, how wave functions, move from place to place, and hopefully gives us an insight into time and to quantum mechanics in the mean time.
Paul Borrill 4:09
Thank you, Sky. So, what’s wrong with time in your opinion?
Sky Nelson-Isaacs 4:16
Well, the Fourier Transforms is a really interesting thing. In physics, we, we know that Fourier transforms are a fundamental part of quantum mechanics, because you know everyone is familiar some extent with the Heisenberg uncertainty principle. This is a trade off in accuracy between position and momentum. But what every physicist knows is that position is represented by sort of a distribution in space or a wave function and momentum is also a distribution in momentum, Fourier transforms of each other. So there’s this fundamental relationship, that is the Fourier transform between these two spaces. You can consider these two real spaces you have the regular space that you’re familiar with this alternate momentum space that you’re maybe not so familiar with. And these are related by the Fourier transform.
But what happens is, we very often use the Fourier transform, like in crystallography, we convert the crystal structure to its frequency space, and we do calculations there all the time. But we very rarely convert the time behavior into the frequency space. And, you know, mathematically, there’s no reason why we shouldn’t, there is a relationship, as an uncertainty relationship between time and energy. If you try to measure something very, very, very brief interval you get a very large spread and energy so there is this trade off. But, like, let’s look at a specific example. In optics, if you want to move an image, like how an image moves from your TV screen to your eyes, because it gets emitted from the TV screen.
And then what we usually do in what’s called Fresnel diffraction is we convert the image on the screen into its frequency representation. In other words, we take the Fourier transform of a two dimensional surface spatially. And then we propagate it, and we propagate it in what’s called the angular spectrum, because the frequency domain you’re now dealing with frequencies, and you’re propagating those. But what’s really important here is this kind of shortcut taken. You don’t convert the time domain, you leave that as it is. So, there’s sort of this crutch where we like to think of time as this spatial thing that moves forward all the moves forward in time, we’re always moving forward in time, and propagate image by evolving in time.
And my question is, you know, from a pure mathematical point of view, if you’re converting the spatial dimensions to the Fourier domain, why wouldn’t you convert the time dimension to the Fourier domain as well. And in doing so, how would you then interpret evolution because evolution usually means evolution over time, so suddenly you’re dealing with representation that has no time in it, because this time is embedded or encoded into the waveforms. So how do you interpret that and most of my work here is actually restating some formalisms that already exist, but interpreting them I think in a way that it makes more of a coherent picture.
Paul Borrill 7:12
Thank you Sky. So tell me, what is a hologram and why is it relevant, in your view?
Sky Nelson-Isaacs 7:20
Well let me describe something about digital audio files first, because it’s really a useful technology to help us understand the purpose of what I’m trying to do. In a digital audio file you have music that plays like a five minute song, you know, it’s a WAV file and uncompressed audio file. And you would convert that file into its frequency representation in order to modify, like the equalization, if you want to change the high end or the low end as everyone’s familiar with. You can convert it to its frequencies and modify those things. But when you do that, you lose any kind of time ordering. So if you tried to play that file on your computer couldn’t do it. It’s the same size data, no data has been lost, but it’s not ordered according to time anymore, it’s ordered according to frequency. And so in order to play it back you have to invert the form back to time.
So what this illustrates is that there is a coherent way of talking about a system that is originally time ordered, but you can encode that time ordering, or hide the time ordering into an alternate representation. So that’s a very useful practical example of what we’re really doing here with the wavefunction that evolves in time. So a hologram, to get to the question. A hologram is essentially capturing the frequency domain. A lot of people think about the important thing about a hologram is, is it captures a three dimensional image, on a two dimensional piece of film. And the way it does that is by, let’s say you have a speck of dust, just to represent like a point particle, and a speck of dust, you shine light on the speck of dust, it bounces off the dust and shines onto a film. And you capture that light on a film you’re going to be trying to capture an image of the speck of dust. And again a speck of dust I’m using just as sort of a point particle to see what’s the simplest possible image you can create. So to create a hologram you shine light on this object, you capture the light that bounces off of it onto the film, and you interfere that light with just a plain beam of laser light coming in. So now you have laser light coming in that’s plain, and laser lights coming in from the object and you interfere them so that you basically take the, you subtract the phases of those two sources of light. And so your film captures a really mathematically accurate representation of what’s, what is in the visual scene around you. And if you, if you take like a rubber ducky and you look at the beak versus the wing, those are just different points that all all emanate light in the same way, and create this much more complicated image of a holographic image of a duck versus just a point of dust.
But nonetheless, the process is pretty straightforward, and it’s capturing the difference in phase on the film. And then when you shine a laser light on that film, without the duck present, do you actually see that the light that passes through the film takes on all the characteristics that it would have had if the duck was there, because the film is actually imparting the right phases, exactly to mimic what you originally had there. So then when you shine the laser on it, and nothing else is there and you actually see the duck floating in space where it originally appeared to have been.
So that’s a hologram, and it’s a very cool technology, but it’s more than just capturing a three dimensional image in the two dimensional piece of film. That’s kind of the hack of it, sort of a technological hack that we can do, but what it really is doing is it’s capturing the frequency domain, as it is. And usually, you know, when we take a photograph, if you just switch gears and think about photographs on your phone, for instance, you’ve got a lens in there, and what the lens is doing is performing a Fourier transform. So it’s taking the light out of the frequency domain, putting it back into spatial resolution, and then, capturing that. So that’s what our eyes do as well. But what we’re doing with the hologram is not converting it to the spatial resolution, we’re leaving it in the frequency domain and that’s what we’re capturing on the film. So it’s a great starting point for thinking about fields and how they can be represented in the frequency domain.
Paul Borrill 11:24
Thank you, so what’s the key insight in your work that leads to something new here?
Sky Nelson-Isaacs 11:33
So all of what I talked about is, is understood, I think generally. One thing I mentioned is that we don’t usually do this kind of treatment for time, we usually leave time as this independent variable. If you look at like a quantum field, the simple version of a quantum field that is an oscillator. You know you you write the frequency portion, you read the spatial portion in terms of the integral over the frequencies which is essentially a Fourier transform in space. So you do a Fourier transform in space but you don’t do a Fourier transform in time. So your field is oscillating in time.
What I’m doing is, I want to get rid of this sense that time is actually happening in an explicit way. And when you encode, also the time domain into its frequencies, you get what is really a map. It’s not a “one to one” map, as I described earlier, it’s a “one to all” map. So every point of the map corresponds to a distributed information throughout throughout the dual space. Every point in the frequency domain, corresponds to something that shows up everywhere in physical space. But this map, although it totally encodes — there’s no information has been lost — it’s not one to one. And in particular, this kind of map can’t change. So now you have a map of reality that cannot change, because how would it change in time? It’s a map in time, and it’s encoding time. So what we’re doing here is by applying this Fourier transform to time as well, just as we would in an audio file, we get rid of the ability to have this thing — this map — change at all. And then we have to think about, okay if my map of reality can’t change, but the particles in reality are moving around, how is it that my map of reality which doesn’t change, because there’s no…I don’t want to add an extra time parameter like a “meta time” so that the map can change…so this map doesn’t change but somehow it encodes particles that are moving around and have equations of motion, that’s the real challenge here. And that leads to a concept of a path as a whole, which maybe we’ll get to shortly, but directly as a corollary that falls out of the principles of treating space and time — the wave function — as essentially signals in space and time, that are Fourier transformed into the frequency domain.
Paul Borrill 14:02
So, let’s get to that right now so what is meant by “paths as a whole”?
Sky Nelson-Isaacs 14:08
Well a simple example is to think about light. And let’s take light traveling from the Sun to the Earth. It takes about eight minutes to do that. And if you think about that journey, it’s a certain distance. And that frequency domain representation if you’re converting both the time and the space domain into the frequency domain, then the representation you get describes the entire path. So, one…the frequency domain representation…So, the folks who might be familiar with engineering are familiar with this principle called the shift property of the Fourier transform, where if you want to move something in space, you multiply it by a phase factor in the frequency domain. And that’s it that’s all we’re doing here. If you’re moving a particle from the Sun to the Earth that corresponds to a phase factor that is being introduced in the, in the frequency domain.
But that phase factor applies to the whole path, the whole journey. And this really important question that John Wheeler brought up comes into play, this delayed choice experiment. I’m not even approaching it from a quantum mechanics point of view, I’m approaching it from a Fourier transform point of view. If the journey from the side of the Earth is represented by a Fourier transform, then that whole journey is encoded into the frequencies. But now let’s say I take satellite in between space in between the Sun and the Earth and I interject it and I capture the light. And I do this sort of at the last minute. Well that’s with a different endpoint, has a completely different Fourier transform.
And so the key here is two paths of different lengths, are going to have completely different Fourier transforms, because even though part of the journey is the same, just like in the hologram, the whole journey is what determines the frequency domain, the patterns in the frequency domain. So if you change any portion of the journey you’ve completely changed the frequency representation. And this is really the key element, so that the journey from the Sun to the satellite is fundamentally different, there’s no portion of that frequency domain that is equal to the journey from the Sun of the Earth.
And so that brings up this really important question of “if the light was already traveling on its way to the earth, and you change its path, well, you’ve just modified the frequency domain that it was already traveling, according to…you’ve modified the map that it was already traveling by.” And so that’s a really difficult interpretational question, how do you modify a journey this already started, but it’s already been described as a whole. And, in short, the conclusion is that there must be a superposition of multiple of these maps, of these frequency domain maps each of which corresponds to the light traveling from the Sun to a different endpoint. And so you recover this notion of superposition, which is very much a part of the Fourier transform, but it’s also a fundamental part of quantum mechanics.
Paul Borrill 17:19
Thank you. Well, this sounds somewhat like the block universe model in special relativity Minkowski spacetime, can you clarify, please?
Sky Nelson-Isaacs 17:29
Yeah, and I’ll say about the last point that it does become hard to justify separating the macro from the micro in this view of paths as a whole and that’s one of the key challenges to face, is how do we understand this, how do we handle…usually we think about quantum mechanics as a microscopic theory, and that’s something that comes into conflict with this theory.
So, let me say about the block universe. The block universe is this idea from relativity theory that space and time are on equal footing as dimensions. And if you want to lay these out visually you can think of it like a block of ice. And one of those dimensions of the ice is time. And so, if you’re traveling through spacetime, it’s like a crack in the block of ice that moves from one point to another, along both space and time. And it’s like a frozen picture of a trajectory. It’s looking at time, essentially from the outside.
Now, what’s what’s interesting about this block universe is that it really explicitly looks like the world we live in, right? You can you can map positions in space and positions in time from the real world, to this block of ice. The difference here, and what we’re talking about in this model is that we’re talking about a “frequency domain block universe”. So things that, as I mentioned, the frequency domain map of something, even if it’s moving, the frequency domain map is fixed, it doesn’t change, because there’s no time in which to mark that change. So, the path, any notion of a path through the frequency domain block universe is not aligned…it doesn’t resemble the physical path that we’re used to in spacetime. And yet, the physical path of a particle is encoded fully into the phase map of the frequencies, and the frequency map — the phase map — is static. So you’ve got this static dataset that gives rise to equations of motion so that particles can move within the physical space. Which I think is really the really fundamental interesting piece here.
Paul Borrill 19:43
So how then can a dynamical description of nature emerge from such a static data set?
Sky Nelson-Isaacs 19:54
Well, first of all think about, they make the basic calculation here — and for the next question I’d like to let Tony asked this question after in the questions about signal processing and I’m going to touch on signal processing a little bit — but for this question around the dynamical description of nature…when someone does a computation of the wave function, moving forward in this particular model, they’re doing a convolution, just like in image processing you’re doing convolutions. So when are they doing that convolution? The data scientist is doing a convolution, sort of in an abstract moment in time, in the physicists moment in time. But the time itself is a dimension that is part of the convolution. So we can’t say, when the physicist does this calculation, it would imply some kind of extra time parameter.
So because of this again I’ll just restate that the field can’t change during a four dimensional convolution. In other words, a four dimensional image transform that includes time — three dimensions of space one dimension of time — can’t happen at a specific time. So, the mathematician’s work is to find out what needs to happen so that the four dimensional field doesn’t change. And here’s really the key. We want the four dimensional field, space and time field, to not change during an interaction. So you convolve it with, you convolve like the incoming wave function with some kind of filter or other particle, and you get a resulting field, a resulting distribution.
But, you need that resulting distribution to be identical to the original distribution because if it wasn’t identical then something would have changed. In other words, it’s like time would have passed, and you can’t have time pass in this abstract way of calculation. So by holding this notion that the field is…that the thing is static, and requiring that it remains so, you then get a bunch of phases that change. What’s happening in the calculation of convolution, is a bunch of phases that are changing for every frequency, and phases changing…and you require those phases to cancel or to vanish — those phase changes. And those phases are specifically described by variables in space and time.
So by making sure that those phase changes disappear, or they cancel, you essentially get things that look like, you know if you’re familiar with high school algebra you have a equation of motion like x equals V T — position equals velocity times time. And let’s say let’s say you’re going 100 miles an hour. Well if you mark the position at 100 miles, and you mark the time at one hour, then x minus VT is going to be zero, because that is the equation of motion of the particle. You travel 100 miles in one hour and your speed is 100 miles per hour, those are going to cancel. And so what we get is that equation of motion that is allowed is precisely the values for which the particles are moving and their motion in space and time is cancelling out.
So the result is that in the block universe, nothing changes. The frequency domain block universe can’t…doesn’t even notice that any changes happen for the particles, because they’ve changed the position, exactly equivalent to how they’ve changed their time based on the speed they’re going. And so all motion cancels out, which is to say that the beautiful thing about this is the following. If you’ve thought about freezing time before, you might have thought about freezing time that’s like taking a snapshot of time, like a frozen image of the world. And that’s not what happens here. When you freeze time here, you allow things to continue moving along what are called “inertial trajectories”, as long as they don’t…a measurement isn’t made. So, the timeless view is actually one in which there’s this actual evolution of time happening where things are moving along their normal paths, kind of like Newton’s first law of motion where an object in motion tends to stay in motion.
So this is really a key conclusion or way of interpreting the data. That what we think of is the simplest possible view is actually one in which things are moving, they’re just not interacting in the usual way, there’s no measurements being made. And so what we call unitary motion is actually associated with a frozen picture of the frequency domain, and it takes us away from this very much more simple notion of freezing time being like everything, everything stops. I’ll leave it there hopefully maybe we can talk more about this in the Q&A.
Paul Borrill 24:36
Sounds good. Okay so what I’ll do is ask Tony to ask this question later on. I’ve got three or four more questions and then we’ll switch to the panel onstage and then I would like to see some people from the audience ask some questions. So my next question then is, relates to interpretations and how does this relate to the many worlds interpretation that’s preferred by Sean Carroll and David Deutsch?
Sky Nelson-Isaacs 24:36
I love this question. I love Sean Carroll, I listen to his podcast Mindscape. It’s a great podcast. What is really important to understand is that this paper is not an interpretation of quantum mechanics. An interpretation of quantum mechanics starts from the same postulates of quantum mechanics, the six postulates or so, that make up quantum mechanics, and then says, given those, those facts as physical facts, how do we interpret what they mean. And I don’t touch on those postulates at all. I don’t bring them up in any way. So I’m not doing quantum mechanics here in the traditional sense.
My postulates are about optics and image processing. So I basically say, the physical wavefunction is a four dimensional wavefunction, a spacetime wavefunction, and it’s got a counterpart, which is a momentum space wavefunction. It’s both the momentum and energy domains. And that these are related by the Fourier transform as one would expect. And I’ve made that statement and say that you really have to be very strict about your treatment and then see what happens. And the second postulate is simply from optics again, that when you interact these two signals or two systems you convolve them together. And from this, you get an equation of essentially evolution that can be reduced to Schrodinger’s equation, and hopefully some other more complicated, like the Dirac and Klein-Gordon, but I haven’t shown those to be the case. I suspect that they are but it would take significant work to do so. So I don’t want to go beyond what I’ve already shown.
But what what I’m doing is not an interpretation of quantum mechanics, it’s a reformulation of wavefunction propagation, and it does reproduce some of the basic ideas in…specifically superposition of states. And in that in that regard it really mirrors the many worlds hypothesis very well. The most relevant area of agreement is the description of this branching waveform, and the relative nature of measurement in that in that wave form. You know, when you measure a system in Many Worlds theory you’re not actually collapsing to that world, but from the observer’s point of view, you’re getting a particular outcome. And a process I call retroactive event determination, which is very similar to the delayed choice that I mentioned earlier by Wheeler, so it’s this notion that histories are flexible, or that the branching wavefunction is essentially flexible to, I would say, influencing branches of the past that haven’t been determined yet. I think that’s supported by Many Worlds and it’s supported by what I’m doing here so there is some some real crossover but again it’s not an interpretation of quantum mechanics in the usual sense.
Paul Borrill 27:47
Thank you. Well it all raises questions in my mind here about how people think about things and not, not necessarily interpretation but this does seem to me like it relates somewhat to the Cramer transactional interpretation. Can you say something about that?
Sky Nelson-Isaacs 28:03
Yeah, so the Cramer interpretation has some interesting aspects, I can point to a couple. In Cramer’s view, the wave function is a function of space and time. It represents real physical waves. And this is contrary to the Copenhagen interpretation where the wavefunction represents something more abstract. In Copenhagen, you can use a wave function to calculate real probabilities, but it’s like a probability wave, it’s not a physical wave like electromagnetism is. So Cramer makes that sort of bold distinction in his view of the transactional interpretation that the wavefunction is physical. But he specifically says there’s a problem when you convert the wavefunction into its frequency domain — which is something you’re allowed to do in physics — the frequency domain wavefunction does not correspond to real physical waves, because he doesn’t really have a way to interpret what that would mean. What is a physical wave in the frequency domain? And that’s exactly what I’m doing, is sort of making sense of waves in spacetime and in the frequency domain, and how they relate to each other.
But I would not agree that the waves are physical waves. I think that what we’re talking about as waves are informational waves that are giving rise to all of space and time and physics. So they’re not made of physical particles or anything like that. So there’s a limitation to his view that is pretty distinct from what comes out of my view.
He does say that, you know, his transactional wave functions are solutions of the Dirac or the Klein-Gordon equation, they don’t actually come out of the Schrodinger equation, because the Schrodinger equation is first order in time which for anyone who’s not a scientist, means that it just has one solution, the forward in time solution. Whereas, the other equations, the more complicated and more accurate ones have both forward in time and reverse time solutions, what are called advanced waves and retarded waves. And so his wave function doesn’t actually come from the Schrodinger equation, it has to come from these more advanced equations, or two different Schrodinger equations, one which is forward in time and one which is explicitly backwards in time. So there is this crossover in the views. But again, he views wavefunctions as real things, and that doesn’t really line up with my view or with, I think the majority of views in quantum mechanics.
Paul Borrill 30:33
Thank you. Beautiful. So, finally, how does this relate to QBism and models such as Rob Spekkens toy models?
Sky Nelson-Isaacs 30:45
QBism, is really interesting to me. Not because I agree with it, let’s just say that, but I really admire the people that were doing work in it. I like David Mermin is someone I followed for a long time. He’s got some really lucid explanations of what the measurement problem really is and what quantum mechanics is trying to tell us. With literally a paper that’s titled “What is quantum mechanics trying to tell us”, so great work there. And the QBism approach is to, I think if I understand it correctly, is to say that wave functions are really a state of knowledge of the observer or the agent. They don’t represent something physical about the world, which sort of takes away the mystique of what quantum mechanics is really trying to tell us. That there isn’t actually…what would you say…there isn’t actually something that the observer is doing to change the physical system, by making an observation, but that the physical…all the observer is doing is updating the knowledge of the physical system. And it explains…some of the results can be explained that way but when you get to Bell’s theorem, which is essentially measuring entanglement, what I understand is the QBism approach has trouble reconciling with entanglement, because entanglement requires this notion of the observer’s choice of measurement operator having an effect on what are the possible outcomes, both for their particle and for any entangled particles. And the details of that really QBism doesn’t have an answer for, as far as I understand, currently.
Paul Borrill 32:18
Thank you. Excellent. Really beautiful description, Sky, thank you so much. What I’d like to do now is move to questions for people that are on the stage. I know that Tony, you’ve got a question you’d like to ask.
So the rules are very simple. This is a very interesting topic, and these issues concerning the nature of time and causality are perhaps among the most challenging and subtle in science and mathematics. And the distinctions between the various models and paradoxes and so on are exceedingly difficult to pinpoint. So our job today is to try to get these issues on the table. And one way to do this is through open mindedness, come from a place of curiosity, rather than defending or refuting a particular argument.
Tony Schultz 33:45
Thanks, I just first want to say thanks to Paul, for starting this club, and making a space for this, this conversation to happen. And, yeah, I wanted to thank Sky. You know first just for this paper. This is a big accomplishment, and it’s very elegant scholarship, so we’ve had some conversations over the past couple of days and I’ve spent some time with this paper, and it’s been a pleasure to engage. You know, for my development, for my time evolution, I’ve spent a lot of time doing signal processing. So I have this breeding as a physicist and then, you know, I became a little bit more technical in working on the computer. And so you know I spend more time doing signal processing now than pure physics. And that language is just really engaging, and I think might be a familiar framework for many of the people listening. I was hoping that you would speak on these fairly technical terms: convolution, impulse response, and transfer function, and how those are used in your concept development, and then just do the hand holding, kind of guide me through how these relate to holography.
Sky Nelson-Isaacs 35:25
I love that question and I think it is a really useful, tangible point of development of the theory, because that is what guided me to making tangible progress in writing these things down. So, the field we’re talking about here is called scalar diffraction theory, developed by Helmholtz, Kirchoff, and some others. And you can find a great introduction to this, a complicated one, but a pretty thorough one in Goodman’s book on Introduction to Fourier Optics, as I’m sure you’ve done this research, Tony, but other folks haven’t had that background. So, what we’re talking about here…the basic equation in my paper is a convolution equation, where you take a four dimensional signal that represents space and time, and you convolve that with some other four dimensional signal, and that’s really just the mathematics that you would use to apply a filter in Adobe Photoshop. If I have an original two dimensional image, and I want to filter it, I apply a low pass filter, or a high pass filter, or a bandpass filter to convolve the two together and I get a resulting image.
And what’s important about a convolution is the convolution theorem, that applies to it, which says that what you’re really doing is converting to the frequency domain, and multiplying. So when you convert a filter to the frequency domain, and then multiply it by your original signal, you’re essentially…one example would be a low pass filter, you’re…you’re masking out all the high frequencies. In other words you’re blanking out or zeroing out all the high frequencies by multiplying by zeros everywhere there’s a high frequency. And that’s in the frequency domain. When you convert it back to the spatial domain, via an inverse Fourier transform, you get a picture that has all the high frequency data removed.
What that looks like physically is all the details are gone, your picture has been blurred, because high frequency data in a picture corresponds to that all the quick variation. So, filters…that’s the way filters work, through convolution. And what you what you typically do is, to understand how your filter is going to work, you break it down into the most basic form, which is called the impulse response. So any kind of camera or digital or analog imaging system will have an impulse response. And that’s the way it responds to the most basic signal. And you convolve your incoming signal with your impulse response, you find out what that looks like, and then you integrate that over the actual filter itself, and it’s like adding up all those point particles to create the overall hologram that I was talking about earlier.
So, what you’re doing here is…in the physics…I’m suggesting the impulse response shows up as well, and the impulse response…when you convert it, you’re convolving it with your signal. So you did the convolution theorem, you’re actually doing that in the frequency domain, that turns into what’s called the transfer function which is how your…the transfer function just describes how your frequencies evolve in the frequency domain. And, in short, the way that this gets written in the paper is this very elegant way of looking at it. Which is part of the appeal is how straightforward this is.
You start with — and this is the basic evolution of a system in this model — you start with your original four dimensional wave function, like space and time. You multiply it by some phase distribution, which is some complicated distribution, that corresponds to what’s called a transmittance function in optics, which is like, just your image, your original acetate or overhead projector. You know like your piece of film that you shine light through, that’s called the transmittance function. That’s the thing you’re trying to change. And that actually correlates to the potential in physics. You multiply your original wave function by this thing that has to do with the potential in the spatial domain, and then you convert the whole thing into the frequency domain. And in the frequency domain, you do the same thing. You multiply by another phase factor, which has to do with the kinetic portion of the Hamiltonian…in my example is momentum squared over 2m, which is just…the Hamiltonian has to do with the energy, the kinetic energy of the system.
So that shows up in the frequency domain, as a phase factor, and then you convert that whole thing back to regular space. And that’s all you do to understand the evolution of the system. And then you make sure that you set the beginning equal to the end, because time can’t actually evolve, and then what you get is some constraints on how the phases must change.
But the point is that how the phase has changed determines the equation of motion. Okay. The point is that all of this is that the original signal, multiplied by a transmittance function, convolved with an impulse response — which has to do with the kinetic energy — and then obtaining the final result in spacetime. So it really is drawn directly from Fourier optics, and the first appendix really just makes that available for folks in the paper. Thanks for that question too, I think it’s great.
And I guess there’s one more point about holograms; that holograms are really described by scalar diffraction theory and Fresnel diffraction, which is a special case that I’m talking about here. And what a hologram is is just capturing the phase differences between the frequencies of a signal and reproducing those for the viewer. So they’re described by exactly the type of physics we’re talking about here.
I will also point out something I didn’t get to mention about a hologram. There is this disconnect between the film itself — when you look at a hologram, the film is kind of invisible, it’s meant to stay in the background and you’re not supposed to see it — and then the image appears in the film as something you can actually see. There’s this direct, exact correlation in the theory between that description of holograms and the physics of the fields. The field or the wavefunction is something that is not visible, you can’t measure it in the background. And the space time event, like a detector capturing a particle, is exactly equivalent mathematically to the holographic image that you see in the film. So there’s this real direct correlation between the spacetime physics and the optics, and holograms.
