In my last post, I looked at how 1920’s quantum physics (“Quantum Mechanics”, or QM) conceives of a particle with definite momentum and completely uncertain position. I also began the process of exploring how Quantum Field Theory (QFT) views the same object. I’m going to assume you’ve read that post, though I’ll quickly review some of its main points.
In that post, I invented a simple type of particle called a Bohron that moves around in a physical space in the shape of a one-dimensional line, the x-axis.
- I discussed the wave function in QM corresponding to a Bohron of definite momentum P1, and depicted that function Ψ(x1) (where x1 is the Bohron’s position) in last post’s Fig. 3.
- In QFT, on the other hand, the Bohron is a ripple in the Bohron field, which is a function B(x) that gives a real number for each point x in physical space. That function has the form shown in last post’s Fig. 4.
We then looked at the broad implications of these differences between QM and QFT. But one thing is glaringly missing: we haven’t yet discussed the wave function in QFT for a Bohron of definite momentum P1. That’s what we’ll do today.
The QFT Wave Function
Wave functions tell us the probabilities for various possibilities — specifically, for all the possible ways in which a physical system can be arranged. (That set of all possibilities is called “the space of possibilities“.)
This is a tricky enough idea even when we just have a system of a few particles; for example, if we have N particles moving on a line, then the space of possibilities is an N-dimensional space. In QFT, wave functions can be extremely complicated, because the space of possibilities for a field is infinite dimensional, even when physical space is just a one-dimensional line. Specifically, for any particular shape s(x) that we choose, the wave function for the field is Ψ[s(x)] — a complex number for every function s(x). Its absolute-value-squared is proportional to the probability that the field B(x) takes on that particular shape s(x).
Since there are an infinite number of classes of possible shapes, Ψ in QFT is a function of an infinite number of variables. Said another way, the space of possibilities has an infinite number of dimensions. Ugh! That’s both impossible to draw and impossible to visualize. What are we to do?
Simplifying the Question
By restricting our attention dramatically, we can make some progress. Instead of trying to find the wave function for all possible shapes, let’s try to understand a simplified wave function that ignores most possible shapes but gives us the probabilities for shapes that look like those in Fig. 5 (a variant of Fig. 4 of the last post). This is the simple wavy shape that corresponds to the fixed momentum P1:
where A, the amplitude for this simple wave, can be anything we like. Here’s what that shape looks like for A=1:
If we do this, the wave function for this set of possible shapes is just a function of A; it tells us the probability that A=1 vs. A=-2 vs. A=3.2 vs. A=-4.57, etc. In other words, we’re going to write a restricted wave function Ψ[A] that doesn’t give us all the information we could possibly want about the field, but does tell us the probability for the Bohron field B(x) to take on the shape A cos(P1 x).
This restriction to Ψ[A] is surprisingly useful. That’s because, in comparing the state containing one Bohron with momentum P1 to a state with no Bohrons anywhere — the “vacuum state”, as it is called — the only thing that changes in the wave function is the part of the wave function that is proportional to Ψ[A].
In other words, if we tried to keep all the other information in the wave function, involving all the other possible shapes, we’d be wasting time, because all of that stuff is going to be the same whether there’s a Bohron with momentum P1 present or not.
To properly understand and appreciate Ψ[A] in the presence of a Bohron with momentum P1, we should first explore Ψ[A] in the vacuum state. Once we know the probabilities for A in the absence of a Bohron, we’ll be able to recognize what has changed in the presence of a Bohron.
The Zero Bohron (“Vacuum”) State
In the last post, we examined what the QM wave function looks like that describes a single Bohron with definite momentum (see Fig. 3 of that post). But what is the QM wave function for the vacuum state, the state that has no Bohrons in it?
The answer: it’s a meaningless question. QM is a theory of objects that have positions in space (or other simple properties.) If there are no objects in the theory, then there’s… well… no QM, no wave function, and nothing to discuss.
[You might complain that the Bohron field itself should be thought of as an “object” — but aside from the fact that this is questionable (is air pressure an object?), the QM of a field is QFT, so taking this route would just prove my point.]
In QFT, by contrast, the “vacuum state” is perfectly meaningful and has a wave function. The full vacuum state wave function Ψ[s(x)] is too complicated for us to talk about today. But again, if we keep our focus on the special shapes that look like cos[P1 x], we can easily write the vacuum state’s wave function for that shape’s amplitude, Ψ[A].
Understanding the Vacuum State’s Wave Function
You might have thought, naively, that if a field contains no “particles”, then the field would just be zero; that is, it would have 100% probability to take the form B(x)=0, and 0% probability to have any other shape. This would mean that Ψ[A] would be non-zero only for A=0, forming a spike as shown in Fig. 6. Here, employing a visualization method I use often, I’m showing the wave function’s real part in red and its imaginary part in blue; its absolute-value squared, in black, is mostly hidden behind the red curve.
We’ve seen a similar-looking wave function before in the context of QM. A particle with a definite position also has a wave function in the form of a spike. But as we saw, it doesn’t stay that way: thanks to Heisenberg’s uncertainty principle, the spike instantly spreads out with a speed that reflects the state’s very high energy.
The same issue would afflict the vacuum state of a QFT if its wave function looked like Fig. 6. Just as there’s an uncertainty principle in QM that relates position and motion (changes in position), there’s an uncertainty principle in QFT that relates A and changes in A (and more generally relates B(x) and changes in B(x).) A state with a definite value of position immediately spreads out with a huge amount of energy, and the same is true for a state with a definite value of A; the shape of Ψ[A] in Fig. 6 will immediately spread out dramatically.
In short, a state that momentarily has B(x) = 0, and in particular A=0, won’t remain in this form. Not only will it change rapidly, it will do so with enormous energy. That does not sound healthy for a supposed vacuum state — the state with no Bohrons in it — which ought to be stable and have low energy.
The field’s actual vacuum state therefore has a spread of values for A — and in fact it is a Gaussian wave packet 
Another way to represent this same wave function involves plotting points at a grid of values for A, with each point drawn in gray-scale that reflects the square of the wave function |Ψ(A)|2, as in Fig. 8. Note that the most probable value for A is zero, but it’s also quite likely to be somewhat away from zero.
But now we’re going to go a step further, because what we’re really interested in is not the wave function for A but the wave function for the Bohron field. We want to know how that field B(x) is behaving in the vacuum state. To gain intuition for the vacuum state wave function in terms of the Bohron field (remembering that we’ve restricted ourselves to the shape cos[P1 x] shown in Fig. 5), we’ll generalize Fig. 8: instead of one dot for each value of A, we’ll plot the whole shape A cos[P1 x] for a grid of choices of A, using gray-scale that’s proportional to |Ψ(A)|2. This is shown in Fig. 9; in a sense, it is a combination of Fig. 8 with Fig. 5.
Remember, this is not showing the probability for the position of a particle, or even that of a “particle”. It is showing the probability in the vacuum state for the field B(x) to take on a certain shape, albeit restricted to shapes proportional to cos[P1 x]. We can see that the most likely value of A is zero, but there is a substantial spread around zero that causes the field’s value to be uncertain.
In the vacuum state, what’s true for a shape with momentum P1 would be true also for any and all shapes of the form cos[P x] for any possible momentum P. In principle, we could combine all of those shapes, for all of the different momenta, together in a much more complicated version of Fig. 9. However, that would make the picture completely unreadable, so I won’t try to do that — although I’ll do something intermediate, with multiple values of P, in later posts.
Oh, and I mustn’t forget to flash a warning: everything I’ve just told you and will tell you for the rest of this post is limited to a child’s version of QFT. I’m only describing what the vacuum state looks like for a “free” (i.e. non-interacting) Bohron field. This field doesn’t do anything except send individual “particles” around that never change or interact with each other. If you want to know more about truly interesting QFTs, such as the ones in the real world — well, expect some things to be recognizable from today’s post, but much of this will, yet again, have to be revisited.
The One-Bohron State
Now that we know the nature of the wave function for the vacuum state, at least when restricted to shapes proportional to cos[P1 x], how does this change in the presence of a single Bohron of momentum P1?
The answer is quite simple: the wave function Ψ(A) changes from 
Notice that the one-Bohron state is clearly distinguishable from the vacuum state; most notably the probability for A=0 is now zero, and its spread is larger, with the most likely values for A now non-zero.
There’s one more difference between these states, which I won’t attempt to prove to you at the moment. The vacuum state doesn’t show any motion; that’s not surprising, because there are no Bohrons there to do any moving. But the one-Bohron state, with its Bohron of definite momentum, will display signs of a definite speed and direction. You should imagine all the wiggles in Fig. 12 moving steadily to the right as time goes by, whereas Fig. 9 is static.
Well, that’s it. That’s what the QFT wave function for a one-Bohron state of definite momentum P1 looks like — when we ignore the additional complexity that comes from the shapes for other possible momenta P, on the grounds that their behavior is the same in this state as it is in the vacuum state.
A Summary of Today’s Steps
That’s more than enough for today, so let me emphasize some key points here. Compare and contrast:
- In QM:
- The Bohron with definite momentum is a particle with a position, though that position is unknown.
- The wave function for the Bohron, spread out across the space of the Bohron’s possible positions x1, has a wavelength with respect to x1.
- In QFT:
- The Bohron “particle” (i.e. wavicle) is intrinsically spread out across physical space [the horizontal x-axis in Figs. 9 and 12] and the Bohron itself has a wavelength with respect to x.
- Meanwhile the wave function, spread out across the space of possible amplitudes A (the vertical axis in Figs. 7a, 8, 10 and 11) does not contain simply packaged information about how the activity in the Bohron field is spread out across physical space x; both the vacuum state and one-Bohron states are spread out, but you can’t just read off that fact from Figs. 8 and 11.
- And note that the wave function has nothing simple to say about the position of the Bohron; after all the spread-out “particle” doesn’t even have a clearly defined position!
Just to make sure this is clear, let me say this again slightly differently. While in QM, the Bohron particle with definite momentum has an unknown position, in QFT, the Bohron “particle” with definite momentum does not even have a position, because it is intrinsically spread out. The QFT wave function says nothing about our uncertainty about the Bohron’s location; that uncertainty is already captured in the fact that the real (not complex!) function B(x) is proportional to a cosine function. Indeed physical space, and its coordinate x, don’t even appear directly in Ψ(A). Instead the QFT wave function, in the restricted form we’ve considered, only tells us the probability that B(x) = A cos[P1 x] for a particular value of A — and that those probabilities are different when there is a single Bohron present (Fig. 12) compared to when there is none (Fig. 9).
I hope you can now start to see why I don’t find the word particle helpful in describing a QFT Bohron. The Bohron does have some limited particle-like qualities, most notably its indivisibility, and we’ll explore those soon. But you might already understand why I prefer wavicle.
We are far from done with QFT; this is just the beginning of our explorations. There are many follow-up questions to address, such as
- Can we put our QFT Bohron into a wave packet state similar to last post’s Fig. 2? What would that look like?
- Do these differences between QM and QFT have implications for how we think about experiments, such as the double-slit experiment or Bell’s particle-pair experiment?
- What do QFT wave functions look like if there are two “particles” rather than just one? There are several cases, all of them interesting.
- How do measurements work, and how are they different, in QM versus QFT?
- What about fields more complicated than the Bohron field, such as the electron field or the electromagnetic field?
We’ll deal with these one by one over the coming days and weeks; stay tuned.
40 Responses
Hi Matt,
Terrific post. As I understand you, Fig. 12 describes several configurations of B(x), shaded according to their measurement-outcome probability. I would have expected any single configuration to cross the x axis at regular intervals. However, the figure makes it look as though B(x) is never equal to 0 for any x (the lines fade to white as they approach the x axis), i.e. it seems as though any single configuration would consist of a series of detached peaks and troughs. Am I looking at this wrong?
Best wishes,
Jon
You’re right to ask the question. There is a question of judgement as to what to draw — after all, the gray curves, black curves and white curves are all superposed here, so I have to decide which ones to plot on top and which ones to plot behind. I chose to emphasize the fact that the amplitude for B(x)=0 is zero, but that does have the negative consequence you mentioned: the white line at B(x)=0 sits on top of the other curves, and so it does make them look disconnected, whereas each and every curve is a connected sine wave, with no detachment.
If I plotted the curves in some other order, then it might be hard to tell that the amplitude for B(x)=0 is zero. But I don’t remember what I tried when I was designing the image. Thanks for the implicit suggestion; I will try it multiple ways next time I return to these questions, and perhaps will revisit the choice I made.
Ah, that makes sense – thank you!
Hi Matt,
I was recommended you by Reddit to answer a question I had, and this blog has helped, but just for completeness I thought I’d pose the question directly.
In short, for the purpose of science communication, what is the “least wrong” way I can draw a quantum particle in QFT? The way that captures the most real life detail I can with the least bad or confusing wrong elements as possible, while still being somewhat simple so I can draw it without requiring hours of code and math. For an electron, how should I draw it? A sphere? A plane wave? A wave packet? A fuzzy sphere? Something else? Does this “least wrong” picture for the electron also hold true for other particles (neutrinos, valence quarks in hadrons, or photons for instance) or do they each need a different looking picture? And how would the “least wrong” picture of antimatter (say a positron) differ from that of an electron?
Idk if this is in your interest to answer this, but in case it is I thought I’d ask. I’ve liked reading through your blog so far, and I’m excited to see if you cover a topic like this even without this comment, or if you cover something else like the weak force, which I’ve struggled with.
In any case, thank you for your work and your time.
This is not an easy question to answer, and in my opinion it does depend on the audience and goals. Could you clarify who you are aiming at and what it is that you’re trying to convey?
Hey thanks for responding.
For the who I’d be aiming at, I’m aiming for science literate and curious people, likely either curious lay people or scientists/students from other fields (like me, Earth Sci major)
I want to try to convey some sense of the main properties of a quantum particle (their wave-like nature, their emergence from an underlying field, the tying of the properties to that field, those properties being associated to discrete energy chunks in that field, which can be added or subtracted from that field). Does this kinda make sense as a goal? I want to sort of draw a very loosely correct image of quantum particles for illustrative devices to get people curious and learning some basics, but can be used as a good jumping off point to go into more in depth, correct, mathematically based explanations.
Sorry if I’m explaining this poorly, I’m just waking up lol
I’ve been trying to think of a way to explain this better. Again, my goal is to try to reach out to science interested and somewhat science literate people who aren’t physics experts already, and as well to have something visually intuitive for my brain to latch onto. My brain likes dumb visuals like that, even if it knows it isn’t 100% literally and the true understanding is only found in the math for instance.
So for example, if someone wanted to draw a free electron as a ball, but was now running into weird misunderstandings of electrons because of that, I want to be able to present a reasonably accurate, QFT inspired drawing of an electron. Something that communicates to that person a more accurate understanding of what the electron actually is, but leave open questions that will be answered by a dive into the math. Ideally this image would introduce the least amount of “BS” as possible (as in deliberately wrong details meant to make the phenomenon more intuitive).
Plus I also am somewhat of an artist, and would like something more visually interesting when drawing diagrams than just using the particle symbol, but I don’t want to use arbitrary abstract shapes or something completely pulled out of thin air, with no real consideration of the actual phenomenon I’m trying to explain. I want something more “interesting” than a particle symbol, but more accurate to our understanding of QFT than a hard ball or abstract shape. Did this make my point seem more clear?
The problem you are up against is not a small one, and I think (based on my own experience trying to do the same thing) that you are probably trying to do too much with one representation.
I have used standing and traveling waves in one dimension — typically broad wave traveling wave-packets and standing wave-packets — as the best representation of “particles” (meaning quanta of fields) with mass. (For particles with no mass, such as photons, there are traveling waves only.) This is what is done in my book and reproduced here in several settings, such as in the series https://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/ and in post-book discussion such as https://profmattstrassler.com/2024/03/12/a-wave-that-stands-on-its-own/ and https://profmattstrassler.com/2024/03/19/yes-standing-waves-can-exist-without-walls/ .
It is possible, but hard, to do something more accurate that really captures field theory. I am still halfway through that process; the first two installments are https://profmattstrassler.com/2025/02/24/the-particle-and-the-particle-part-1/ and https://profmattstrassler.com/2025/02/25/the-particle-and-the-particle-part-2/, see especially Figs. 10-12 . I’ll do more intuitive wave packets sometime soon, but have to finish the double-slit experiment first.
Thank you so much, I’ll look into those posts you provided. I’m excited to see you tackle that in the future and you’re current posts are really helping me better understand what superposition actually means without the gobbledygook.
Idk if you consider audience question for what you cover in the future, but I would really like a deep dive into the weak force at some point, since I’ve seen lots of conflicting answers online about the force being “attractive” or “repulsive” and what charge/charges dictate interactions via the weak force.
For instance, the weak “charges” involved in elastic (Z Boson) vs inelastic (W Boson) scattering of an electron and neutrino.
The weak nuclear interaction is complicated, but I have written about it occasionally. Like electromagnetism, it can be both attractive and repulsive, and it is also transformative. The strong nuclear interaction is also attractive, repulsive and transformative, but much less obviously.
If you use the search function on this site, you should find some articles. For instance, https://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-astonishing-standard-model/why-the-weak-nuclear-force-is-short-range/ , https://profmattstrassler.com/2022/08/01/celebrating-the-standard-model-how-we-know-quarks-come-in-three-colors/ , https://profmattstrassler.com/2024/11/20/celebrating-the-standard-model-the-magic-angle/ , https://profmattstrassler.com/2024/12/19/the-standard-model-more-deeply-the-magic-angle-nailed-down/
In quantum theory one often talks about the collapse of the wave function.
I’m curious to know if there will also be a collapse of the wavicle in addition to the collapse of the wave function. My gut feeling is there shouldn’t be because QFT should be local, but I can’t really visualize how it is going to work.
I’m guessing ( and hoping ) this will all become clearer to me in your upcoming posts about the double slit experiment in QFT.
“One often talks…”; I would say, “One often hears talk.”
Wavicles are physical objects and do not collapse, any more than sound waves collapse.
Wave functions in any quantum system can be viewed, from a certain perspective, as “collapsing”, but we’ll see exactly what this means soon and you can judge for yourself. We won’t need anything as complicated as the double-slit experiment to look into this.
Looking at Table 1 of your previous post I am confused. How can you reconcile these two views of fundamental physics? I would hope there is some way to “translate” between these two views, as there are translations between relativity and classical physics, and a translation between the path integral formulation of QM and the Lagrangian formulation of classical physics. Is there such a translation? They seem so different? Will the double slit explanation of QFT clear things up for me?
Good question. The issue of how to translate is important, and I will at some point explain it; but I haven’t worked out pedagogically the best strategy for this yet. The point is that going from the Schrodinger wave function for a field in a one-particle state to the Schrodinger equation for that particle alone in a setting with low velocities (so that Einstein’s relativity isn’t necessary) isn’t a trivial step; it requires some math, some approximations, and an introduction of a space of possibilties that does not arise directly from the space of possibilities for QFT. Anyway, I have work to do before I can properly describe how this happens. The double-slit experiment is too complicated to explain this issue, but we won’t need anything as complicated as that to see how it works; a particle in empty space is probably enough.
May I be so bold to suggest that you have enough material here to warrant another full book. A sequel to the “Waves” book would be great, though this subject does seem a bit more mathematical than the “Waves” book.
It may happen…
My bookshelf is waiting… 🙂
Wow, I’m trying to get through this and the previous post to get a coherent picture . A question : in qm we know that E=hf, but we know also that in qm ‘particles’ ARE particles so the frequency that appear in the equation is related to the wave function that describe the particle itself? Thanks Matt
Yes, in the Schroedinger viewpoint the frequency comes from the Hamiltonian evolution, and the Hamiltonian gives you the energy. So the relation E=hf is baked in to quantum physics of any form.
Thanks Matt. let me open a more philosophical window: given a kind of ‘instability’ of the vacuum state, can this somehow be related to the fact that ‘something exists instead of nothing’?
And if everything is somehow related to fields, can space and time be ripples of some field? Thank you
The answer to your last question is no, but that’s because you’re using some wrong categories; it’s too long for me to get into here. You should probably read my book, because I spent many pages on these issues.
For your first question: I don’t know what you mean by “instability of the vacuum state”. Usually that term is loose langauge meaning that the state in question is not actually the vacuum state; it’s a “false vacuum” that will eventually decay to the “true vacuum state” where there will still be something, not nothing. In fact the decay will release enormous amounts of energy and there will be lots and lots of something after the decay. But maybe you meant something else…?
thanks. you wrote: In short, a state that momentarily has B(x) = 0, and in particular A=0, won’t remain in this form. Not only will it change rapidly, it will do so with enormous energy. That does not sound healthy for a supposed vacuum state — the state with no Bohrons in it — which ought to be stable and have low energy.I understood it as if the state of absence of bohrons (nothingness) was less probable than a situation in which bohrons (something) appear, hence the assumption that ‘something instead of nothing’ is more probable…
“Something instead of nothing” would indeed be more probable if I tried to make a state that had B(x) exactly equal to 0. But that’s *not* the vacuum state; it’s not even a false vacuum state. It is a combination (“superposition”) of extremely highly excited states, in which there is a huge amount of energy and lots going on, with many Bohrons flying everywhere.
The vacuum state, where there is nothing — no Bohrons — is the one in which A is not generally zero but instead has probability to be found over a range, and [when we account not only for one possible shape but all of them] B(x) has almost zero probability of being exactly zero.
But the fact that B(x) is non-zero in the vacuum does not mean that there are any Bohrons — any something — in the vacuum. B(x) is the field. Bohrons are the somethings, and they are resonant ripples in the field; general disturbances in the field, of random shape, are not somethings.
OK thanks I think I got the point
the relation between the physical referent and a higher dimensional reference brings us to weak theta as achtually a member of that Theta Vaccua higher dimensional resource toward vaccua, so is it a Yang Mills application that the higher dimension flipping the third form the basis of anti wavicles as a consequence of the successive mapping of branes in the winding number?
There was a nice paper a few years ago from Helmut Linde with visuals along these lines: https://arxiv.org/abs/1907.11311
How scandalous is this series of articles going to get for particle aficionados? Are you going to make people confront the Hegerfeldt/Reeh-Schlieder theorems against particle localization? Bogoliubov transformations in general curved backgrounds? Haag’s theorem?
Seems I largely rediscovered his method independently.
There won’t be anything scandalous.
All the last three figures are labeled “Figure 10”.
Thanks – fixed!
Dr.Strassler:
I had always thought it odd to think of particles as solid little spheres, localized at a point. The Greeks thought that the atom was the smallest constituent of matter (I believe the word Atom in Greek means indivisible) Then it was discovered that the atom had smaller components, electrons, protons, neutrons… then it was discovered that protons and neutrons had smaller components, Quarks. At some point, there can no smaller “solid” component, the smallest component must be something totally different, like an undulating blob of energy. This is why I think that QFT makes the most sense. Now the question: If the blob of energy has a meaningful spread of 5 inches, (in other words 99% of the energy is spread over 5 inches) is that considered localized? I guess what I’m asking, from a QFT standpoint, what is considered localized?
Localized is a relative term, so indeed, I’ve been slippery about defining it. From the QFT standpoint, the question can only be answered relative to something else. When you include gravity, there’s a natural comparison: the Planck length. https://en.wikipedia.org/wiki/Planck_units
Phew, that is quite an article!
Could you help me relate some of this to the little that I’ve read in the textbooks?
In doing s(x) = A cos (P1 x) you’re doing a momentum space mode expansion of B, a Fourier transform. Is that right?
Is psi[s] the vacuum expectation value: ? (Although I thought that really would be zero for all fields except the Higgs?)
Just to give you a little background. I did maths and physics as an undergrad about 50 years ago, but haven’t done any since. I’ve been trying, not very successfully, to read up on this stuff as a retirement hobby. I really appreciate your posts and am hoping that I’ll finally be able to make some sense of QFT.
Aghhh – the platform’s deleting some of what I wrote as it’s interpreting it as html. Don’t know how to get round that.
Yeah, it doesn’t like open and closed brackets; there’s a way around it, see https://www.w3schools.com/html/html_entities.asp
First question: Yes.
Second question: No! Superimportant. Do not confuse a wave function with a field. Psi is a wave function, not a field. A field is a function of x, of physical space. A QFT wave function is a function of possible shapes. There is absolutely no relation whatsoever, other than that people weren’t creative enough with their lettering choices.
A vacuum expectation value of a field would be B(x) = constant (on average, also written <B(x)> = constant). Absolutely nothing to do with a wave function of a shape, which is about probabilities for possibilities. Super, super important to get this straight… think it over hard!
Thanks for that. Clearly I need to reread more carefully, but it’s early evening here in the UK and I’m probably not thinking at my best. At least, that’s my excuse.
Couple of more questions.
The condition that the values of “A” form a Gaussian. Although that is necessary for the reasons you outline, nevertheless, this isn’t a calculated condition. It’s being imposed separately. Is that right?
Could you give me a hint about what the formula is for psi? (Apologies if this is obvious.)
First question: in this child QFT — noninteracting fields — it is a calculation, not an imposition. The same is true for the one-Bohron state. I’ll show the calculation soon enough.
Second question: the full wave function looks something like this:
First take B(x) and fourier transform x to p, so that B(x) = \integral dp e^{ipx} A(p) ; I’m dropping all hbar’s and c’s from my expressions to keep them short.
Then Psi[B(x)] can be rewritten in terms of Psi[A(p)], which will give us the probability that the mode with wavelength 2 pi/p has amplitude A(p). (In this post we’ve singled out A(p) for k = p_1 and called it just “A”.)
Then for the vacuum state, Psi[A(p)] = Exp[-\integral dp f(p) A(p)^2/2] , where f(p) is some function that depends on the momentum in a simple way but which I won’t try to reconstruct now. The important thing is that it has a Gaussian dependence on A(p) for each p.
Thanks. Those functions for the spread of A(p1) look suspiciously like the wavefunctions for the first two states of the harmonic oscillator in traditional QM.
Indeed they do; even in classical physics, wave equation solutions are oscillatory, just as for a particle in a classical harmonic oscillator. So this is not an accident.
Although out of depth here, I appreciate the faculties of intelligence and its attendance product (thought). When I say Intelligence, I mean MIND which is external to the decaying and dying brain. So in the context you share, it is obvious you are expanding your horizons with the creativity that is also non empirical, a gift you inherited but also able to amplify. Keep up the good work where the unknown becomes known Kind regards Joseph