Why do I find the word particle so problematic that I keep harping on it, to the point that some may reasonably view me as obsessed with the issue? It has to do with the profound difference between the way an electron is viewed in 1920s quantum physics (“Quantum Mechanics”, or QM for short) as opposed to 1950s relativistic Quantum Field Theory (abbreviated as QFT). [The word “relativistic” means “incorporating Einstein’s special theory of relativity of 1905”.] My goal this week is to explain carefully this difference.
The overarching point:
- in QM, an electron really is a particle, at least to the limited extent that QM can handle that notion;
- in QFT, an electron is at best a “particle”, with the word in quotation marks
- (and I personally prefer the term wavicle.)
I’ve discussed this to some degree already in my article about how the view of an electron has changed over time, but here I’m going to give you a fuller picture. To complete the story will take two or three posts, but today’s post will already convey one of the most important points.
There are two short readings that you may want to dofirst.
- The first is an article about the distinction between physical space (within which all objects actually are located) and the abstract space of possibilities (which consists of all ways that the objects in a physical system could be arranged in physical space.) This issue, essential in quantum physics of any type, will be crucial below.
- The second is a short section of a blog post that explains how QM views an isolated object in a state of definite momentum — i.e. something whose motion (both speed and direction) is precisely known, and whose position is therefore completely unknown, thanks to Heisenberg’s uncertainty principle.
I’ll will review the main point of the second item, and then I’ll start explaining what an isolated object of definite momentum looks like in QFT.
Removing Everything Extraneous
First, though, let’s make things as simple as possible. Though electrons are familiar, they are more complicated than some of their cousins, thanks to their electric charge and “spin”, and the fact that they are fermions. By contrast, bosons with neither charge nor spin are much simpler. In nature, these include Higgs bosons and electrically-neutral pions, but each of these has some unnecessary baggage. For this reason I’ll frame my discussion in terms of imaginary objects even simpler than a Higgs boson. I’ll call these spinless, chargeless objects “Bohrons” in honor of Niels Bohr (and I’ll leave the many puns to my readers.)
For today we’ll just need one, lonely Bohron, not interacting with anything else, and moving along a line. Using 1920s QM in the style of Schrödinger, we’ll take the following viewpoints.
- A Bohron is a particle and exists in physical space, which we’ll take to be just a line — the set of points arranged along what we’ll call the x-axis.
- The Bohron has a property we call position in physical space. We’ll refer to its position as x1.
- For just one Bohron, the space of possibilities is simply all of its possible positions — all possible values of x1. [See Fig. 1]
- The system of one isolated Bohron has a wave function Ψ(x1), a complex number at each point in the space of possibilities. [Note it is not a function of x, the points in physical space; it is a function of x1, the possible positions of the Bohron.]
- The wave function predicts the probability of finding the Bohron at any selected position x1: it is proportional to |Ψ(x1)|2, the square of the absolute value of the complex number Ψ(x1).
A QM State of Definite Momentum
In a previous post, I described states of definite momentum. But I also described states whose momentum is slightly less definite — a broad Gaussian wave packet state, which is a bit more intutive. The wave function for a Bohron in this state is shown in Fig. 2, using three different representations. You can see intuitively that the Bohron’s motion is quite steady, reflecting near definite momentum, while the wave function’s peak is very broad, reflecting great uncertainty in the Bohron’s position.
- Fig. 2a shows the real and imaginary parts of Ψ(x1) in red and blue, along with its absolute-value squared |Ψ(x1)|2 in black.
- Fig. 2b shows the absolute value |Ψ(x1)| in a color that reflects the argument [i.e. the phase] of Ψ(x1).
- Fig. 2c indicates |Ψ(x1)|2, using grayscale, at a grid of x1 values; the Bohron is more likely to be found at or near dark points than at or near lighter ones.
For more details and examples using these representations, see this post.
To get a Bohron of definite momentum P1, we simply take what is plotted in Fig. 2 and make the broad peak wider and wider, so that the uncertainty in the Bohron’s position becomes infinite. Then (as discussed in this post) the wave function for that state, referred to as |P1>, can be drawn as in Fig. 3:
In math, the wave function for the state at some fixed moment in time takes a simple form, such as
![\Psi_{|{P_1}\rangle} (x_1) = \cos [P_1 x_1] + i \sin [P_1 x_1]](https://s0.wp.com/latex.php?latex=%5CPsi_%7B%7C%7BP_1%7D%5Crangle%7D+%28x_1%29+%3D+%5Ccos+%5BP_1+x_1%5D+%2B+i+%5Csin+%5BP_1+x_1%5D+&bg=ffffff&fg=000&s=0&c=20201002)
where i is the square root of -1. This is a special state, because the absolute-value-squared of this function is just 1 for every value of x1, and so the probability of measuring the Bohron to be at any particular x1 is the same everywhere and at all times. This is seen in Fig. 3c, and reflects the fact that in a state with exactly known momentum, the uncertainty on the Bohron’s position is infinite.
Let’s compare the Bohron (the particle itself) in the state |P1> to the wave function that describes it.
- In the state |P1>, the Bohron’s location is completely unknown. Still, its position is a meaningful concept, in the sense that we could measure it. We can’t predict the outcome of that measurement, but the measurement will give us a definite answer, not a vague indefinite one. That’s because the Bohron is a particle; it is not spread out across physical space, even though we don’t know where it is.
- By contrast, the wave function Ψ(x1) is spread out, as is clear in Fig. 3. But caution: it is not spread out across physical space, the points of the x axis. It is spread out across the space of possibilities — across the range of possible positions x1. See Fig. 1 [and read my article on the space of possibilities if this makes no sense to you.]
- Thus neither the Bohron nor its wave function is spread out in physical space!
We do have waves here, and they have a wavelength; that’s the distance between one crest and the next in Fig. 3a, and the distance beween one red band and the next in Fig. 3b. That wavelength is a property of the wave function, not a property of the Bohron. To have a wavelength, an object has to be wave-like, which our QM Bohron is not.
Conversely, the Bohron has a momentum (which is definite in this state, and is something we can measure). This has real effects; if the Bohron hits another particle, some or all of its momentum will be transferred, and the second particle will recoil from the blow. By contrast, the wave function does not have momentum. It cannot hit anything and make it recoil, because, like any wave function, it sits outside the physical system. It merely describes an object with momentum, and tells us the probable outcomes of measurements of that object.
Keep these details of wavelength (the wave function’s purview) and the momentum (the Bohron’s purview) in mind. This is how 1920’s QM organizes things. But in QFT, things are different.
First Step Toward a QFT State of Definite Momentum
Now let’s move to quantum field theory, and start the process of making a Bohron of definite momentum. We’ll take some initial steps today, and finish up in the next post.
Our Bohron is now a “particle”, in quotation marks. Why? Because our Bohron is no longer a dot, with a measurable (even if unknown) position. It is now a ripple in a field, which we’ll call the Bohron field. That said, there’s still something particle-like about the Bohron, because you can only have an integer number (1, 2, 3, 4, 5, …) of Bohrons, and you can never have a fractional number (1/2, 7/10, 2.46, etc.) of Bohrons. This feature is something we’ll discuss in later posts, but we’ll just accept it for now.
As fields go, the Bohron field is a very simple example. At any given moment, the field takes on a value — a real number — at each point in space. Said another way, it is a function of physical space, of the form B(x).
Very, very important: Do not confuse the Bohron field B(x) with a wave function!!
- This field is a function in physical space (not the space of possibilities). B(x) is a function of physical space points x that make up the x-axis, and is not a function of a particle’s position x1, nor is it a function of any other coordinate that might arise in the space of possibilities.
- I’ve chosen the simplest type of QFT field: B(x) is a real number at each location in physical space. This is in contrast to a QM wave function, which is a complex number for each possibility in the space of possibilities.
- The field itself can carry energy and momentum and transport it from place to place. This is unlike a wave function, which can only describe the energy and momentum that may be carried by physical objects.
Now here’s the key distinction. Whereas the Bohron of QM has a position, the Bohron of QFT does not generally have a position. Instead, it has a shape.
If our Bohron is to have a definite momentum P1, the field must ripple in a simple way, taking on a shape proportional to a sine or cosine function from pre-university math. An example would be:
![B(x) = A \cos [P_1 x] \ ,](https://s0.wp.com/latex.php?latex=B%28x%29+%3D+A+%5Ccos+%5BP_1+x%5D+%5C+%2C+&bg=ffffff&fg=000&s=0&c=20201002)
where A is a real number, called the “amplitude” of the wave, and x is a location in physical space.
At some point soon we’ll consider all possible values of A — a part of the space of possibilities for the field B(x) — so remember that A can vary. To remind you, I’ve plotted this shape for A=1 in Fig. 4a and again for A=-3/2 in Fig 4b.
Initial Comparison of QM and QFT
At first, the plots in Fig. 4 of the QFT Bohron’s shape look very similar to the QM wave function of the Bohron particles, especially as drawn in Fig. 3a. The math formulas for the two look similar, too; compare the formula after Fig. 3 to the one above Fig. 4.
However, appearances are deceiving. In fact, when we look carefully, EVERYTHING IS COMPLETELY DIFFERENT.
- Our QM Bohron with definite momentum has a wave function Ψ(x1), while in QFT it has a shape B(x); they are functions of variables which, though related, are different.
- On top of that, there’s a wave function in QFT too, which we haven’t drawn yet. When we do, we’ll see that the QFT Bohron’s wave function looks nothing like the QM Bohron’s wave function. That’s because
- the space of possibilities for the QM wave function is the space of possible positions that the Bohron particle can have, but
- the space of possibilities for the QFT wave function is the space of all possible shapes that the Bohron field can have.
- The plot in Fig. 4 shows a curve that is both positive and negative but is drawn colorless, in contrast to Fig. 3b, where the curve is positive but colored. That’s because
- the Bohron field B(x) is a real number with no argument [phase], whereas
- the QM wave function Ψ(x1) for the state of definite momentum has an always-positive absolute value and a rapidly varying argument [phase].
- The axes in Fig. 4 are labeled differently from the axis in Fig. 3. That’s because (see Fig. 1)
- the QFT Bohron field B(x) is found in physical space, while
- the QM wave function Ψ(x1) for the Bohron particle is found in the particle’s space of possibilities.
- The absolute-value-squared of a wave function |Ψ(x1)|2 is interpreted as a probability (specifically, the probability for the particular possibility that the particle is at position x1. There is no such interpretation for the square of the Bohron field |B(x)|2. We will later find a probability interpretation for the QFT wave function, but we are not there yet.
- Both Fig. 4 and Figs. 3a, 3b show curves with a wavelength, albeit along different axes. But they are very different in every sense
- In QM, the Bohron has no wavelength; only its wave function has a wavelength — and that involves lengths not in physical space but in the space of possibilities.
- In QFT,
- the field ripple corresponding to the QFT Bohron with definite momentum has a physical wavelength;
- meanwhile the QFT Bohron’s wave function does not have anything resembling a wavelength! The field’s space of possibilities, where the wave function lives, doesn’t even have a recognizable notion of lengths in general, much less wavelengths in particular.
I’ll explain that last statement next time, when we look at the nature of the QFT wave function that corresponds to having a single QFT Bohron.
A Profound Change of Perspective
But before we conclude for the day, let’s take a moment to contemplate the remarkable change of perspective that is coming into our view, as we migrate our thinking from QM of the 1920s to modern QFT. In both cases, our Bohron of definite momentum is certainly associated with a definite wavelength; we can see that both in Fig. 3 and in Fig. 4. The formula for the relation is well-known to scientists; the wavelength λ for a Bohron of momentum P1 is simply
where h is Planck’s famous constant, the mascot of quantum physics. Larger momentum means smaller wavelength, and vice versa. On this, QM and QFT agree.
But compare:
- in QM, this wavelength sits in the wave function, and has nothing to do with waves in physical space;
- in QFT, the wavelength is not found in the field’s wave function; instead it is found in the field itself, and specifically in its ripples, which are waves in physical space.
I’ve summarized this in Table 1.
Let me say that another way. In QM, our Bohron is a particle; it has a position, cannot spread out in physical space, and has no wavelength. In QFT, our Bohron is a “particle”, a wavy object that can spread out in physical space, and can indeed have a wavelength. (This is why I’d rather call it a wavicle.)
[Aside for experts: if anyone thinks I’m spouting nonsense, I encourage the skeptic to simply work out the wave function for phonons (or their counterparts with rest mass) in a QM system of coupled balls and springs, and watch as free QFT and its wave function emerge. Every statement made here is backed up with a long but standard calculation, which I’m happy to show you and discuss.]
I think this little table is deeply revealing both about quantum physics and about its history. It goes a long way toward explaining one of the many reasons why the brilliant founding parents of quantum physics were so utterly confused for a couple of decades. [I’m going to go out on a limb here, because I’m certainly not a historian of physics; if I have parts of the history wrong, please set me straight.]
Based on experiments on photons and electrons and on the theoretical insight of Louis de Broglie, it was intuitively clear to the great physicists of the 1920s that electrons and photons, which they were calling particles, do have a wavelength related to their momentum. And yet, in the late 1920s, when they were just inventing the math of QM and didn’t understand QFT yet, the wavelength was always sitting in the wave function. So that made it seem as though maybe the wave function was the particle, or somehow was an aspect of the particle, or that in any case the wave function must carry momentum and be a real physical thing, or… well, clearly it was very confusing. It still confuses many students and science writers today, and perhaps even some professional scientists and philosophers.
In this context, is it surprising that Bohr was led in the late 1920s to suggest that electrons are both particles and waves, depending on experimental context? And is it any wonder that many physicists today, with the benefit of both hindsight and a deep understanding of QFT, don’t share this perspective?
In addition, physicists already knew, from 19th century research, that electromagnetic waves — ripples in the electromagnetic field, which include radio waves and visible light — have both wavelength and momentum. Learning that wave functions for QM have wavelength and describe particles with momentum, as in Fig. 3, some physicists naturally assumed that fields and wave functions are closely related. This led to the suggestion that to build the math of QFT, you must go through the following steps:
- first you take particles and describe them with a wave function, and then
- second, you make this wave function into a field, and describe it using an even bigger wave function.
(This is where the archaic terms “first quantization” and “second quantization” come from.) But this idea was misguided, arising from early conceptual confusions about wave functions. The error becomes more understandable when you imagine what it must have been like to try to make sense of Table 1 for the very first time.
In the next post, we’ll move on to something novel: images depicting the QFT wave function for a single Bohron. I haven’t seen these images anywhere else, so I suspect they’ll be new to most readers.
27 Responses
It would help me to get a description of how particles and anti particles are mapped to wavicles as wavicles and anti wavicles” … the control and behaviour axes of mathematics seem to carry the dichotomy of the two physics 20s and 50s era you describe as per also Langlands automorphic and Spectral as dichotomy informing BLK structure towards mapping for example higher dimension such as string relative lower CFt Dimensioning and here returning to the simple course you chart before it gets into “prime gas” territory … but pre-broaching that a bit… it has been said that relativity tensors cover an underlying 16 equation and along with wavicle anti wavicle perhaps this is something you could comment on regarding as it were the flux field toward flux tubes as they may co-define?
To me the key distinction is that QFT deals with number operators that create or destroy particles while in QM the number of particles is assumed to be fixed.
You are thinking of creation and annihilation operators. Number operators just count particles, and often don’t exist in interacting field theory.
True. My mistake.
As atonement here is a Physics parody courtesy of ChatGPTb of Tombstone Blues by Bob Dylan:
I’m standing here with Heisenberg,
And he says, “Where you going, kid?
Uncertainty principle got you in a bind?”
I say, “I can’t measure this damn thing,
I got waves and particles, it’s all so intertwined!”
He laughs and hands me a photon,
And I’m caught in the middle of a wavefunction.
Chorus:
Well, it’s the physics tombstone blues,
Where gravity bends and light will lose,
Einstein’s laughing, Schrödinger too,
And all we can do is try to get through.
What a world this is becoming…
I’m loving this series of articles. One thing I’m hoping to understand is if the QFT view leads to a different explanation of the double-slit interference pattern to the explanation you get under the QM view. My hunch is that it’s still due to interference in the wave function over the “space of possibilities”, but then I’m left wondering what are the wave properties of a wavicle that are observable. e.g. do wavicles diffract and interfere?
You are touching on a point that I do plan to address soon. Indeed, in QFT the new thing is that there is both interference of fields in physical space and interference of possibilities in the space of possibilities, and these can be going on simultaneously. I myself have still not figured out the best way to explain and illustrate this, but I’m confident that my disguised double-slit experiment will offer some opportunities.
I don’t think I have ever seen a proper comparison between QM and QFT, contrasting their particles and wave functions, like this for laypeople like me. Fantastic!
Regardig your “aside to experts”: Is this way to present it not already well known among experts?
Well, there are a lot of different types of experts. In many areas of physics and chemistry and astronomy, people who are experts in how to use QM (not necessarily in how to think about it conceptually) may not be experts in QFT; indeed they may not even be familiar with it, and if they are, they may not have learned its relativistic version. Experimental atomic physicists, for instance, do not need QFT. Many philosophers of physics don’t know QFT. In addition, many people learn QFT only enough to use it technically, but really thinking through its conceptual foundations is not something covered in most QFT classes. So this comparison of QM with QFT will not be familiar to them.
As for QFT experts themselves, who are mostly theoretical physicists in particle physics, string theory and solid-state physics: I imagine that a substantial minority understands this very well, with the majority never having needed to think this through for the purpose of pedagogy. Most people probably understand the point about the wavelength moving from the wave function in QM to the “particle” in QFT, but haven’t really had a reason to think about how to explain it to a non-expert. A nice example is my friend David Tong, who writes wonderful lectures on QFT and does describe some of these issues, including the calculation described in my comment. He certainly understands these points, but still, he doesn’t go into this kind of careful detail, not because he can’t but because he has other priorities in his class.
As for the specific way I’m presenting it, I have never seen it done, but I would be surprised if it truly has never been done before. QFT has been around for 75 years.
Alright, I admit it: You are starting to get into some interesting turf!
🙂 Just getting started…
Dr.Strassler:
when you say “ripple” I visualize something like a wave pulse. A wave pulse is not a periodic sin or cosine wave, although it is slightly “spread out” with the bulk of the pulse being somewhat “localized” around some position, with the tails training off asymptotically towards zero. In principle, the pulse does not have a wavelength proper, although I have seen a definition of what can be considered a wavelength in a pulse. So, with that said, is a “ripple” in QFT somewhat like a wave pulse?
The word “ripple” is indeed ambiguous. The answer to your question is “no”, an electron can be anything between a pulse and a sine wave, or even far more complicated. The problem is that there is no word in the English language for such a thing.
The next two blog posts on the Particle and “Particle” will give you some sense of this.
Dr.Strassler:
Very much looking forward to additional posts. One additional question:
Wave Packet, is a term generally associated with QM
Is Wave Packet still a “valid term” in QFT?
Wave packets exist in most theories with waves, including pre-quantum waves (e.g. sound waves), general relativity, QM, QFT and beyond. Nothing quantum about them per se.
We’ll see wave packets in QFT, with some difficulty, soon.
So, in classical logic, one distinguishes between sortals and mass terms. Sortals are “individuals” as we discern from, say, proper names. Men are sortals by virtue of using ‘Socrates’ to speak of an individual man. By contrast, “water” is a mass term which must involve an auxiliary term indicating units. That is, we speak of “molecules of water.”
It would seem that the quantum fields in which we are interested are comparable with mass terms. Our sense of units seems to be “disturbances in the field.” Is this meant to be synonymous with ‘wavicle’?
I just want to be clear that “wavicle” should not be thought of as a shortened form for “wavicle field.”
For what this is worth, one of the welcome features of your approach is that you are seem to be trying to emphasize the character of functionals on a function space in order to clarify how to think of the field theory different from 1920’s quantum mechanics.
There is no such thing as a “wavicle field”. A wavicle is an object that can be localized, while that is not true of a field.
The relation between fields and wavicles is not one of sortals/mass-terms. A wavicle is an individual object that is a particular type of disturbance in a field, satisfying certain conditions.
“Disturbances in the field” is too broad; many disturbances do not satisfy the conditions that would make them a wavicle. A wavicle is a resonant phenomenon, while a general disturbance is not. [“Virtual particles” are general disturbances, “real particles” are wavicles, which satisfy a relation between their energy and momentum that general disturbances do not.]
We’ll see wavicles (“particles” in quotes) in the next couple of posts. And yes, we’ll see a piece of a functional on function space in an actual picture, hopefully a helpful one. I haven’t decided yet how to approach “virtual particles”, which (as I always emphasize) aren’t particles in any sense of the word.
Thank you for your precise answer.
With regard to virtual particles, logical contexts use the expression “well-formed” to differentiate expressions “meaningful to the language” from arbitrary expressions formed from concatenating symbols. It seems that the on-shell/off-shell distinction may serve a similar purpose in this context.
There are probably better approaches for your readers. But, I have run across papers calling for virtual particles to have the same ontological status as wavicles. By comparison, the most ardent denials of this position seem to invoke the on-shell/off-shell distinction. So, right or wrong, I have adopted this to my knowledge base until I have some better demarcation.
Thanks again.
Calculations done with “virtual particles” can be done in other ways without ever invoking the notion of virtual particles. The same is not true for wavicles; these are objects that one observes in experiments, and no calculation can undo what is seen in a particle detector. I’ll leave it at that for now.
One more terrific post, Matt. Can we say that the famous wave-particle dualism is a myth (at least in 1920 because in qft the problem does not arise beacuse we don’t have particles) because we are basically talking about two different things: on the one hand we have a real particle that lives in physical space and therefore cannot have wave characteristics, and on the other hand we have a wave function that lives in the space of possibilities, which is not physical space, and which has characteristics typical of a wave ? Thanks
I would say something like that, yes; the wave-particle question was very confusing in the 1920s, and the way it gets resolved has a lot to do with the way special relativity works, so you really have to understand QFT before you can understand what is wave-like and particle-like about electrons.
Dear Matt,
this is a real eye opener for a retired Quantum Chemistry student like me. Bravo, and thanks.
The clear separation between 1-D physical space x and 2D-probability space (x1,p1) rises a question in my mind:
normalisation of anything spread out over physical space is just ‘conservation of stuff’.
But I’m unsure how normalisation of probabilities works out over probability space if either x1 or p1 change in time. If x1 broadens with time, does p1 get narrower? cfr fig.7 https://i0.wp.com/profmattstrassler.com/wp-content/uploads/2025/02/X0p_state.gif?ssl=1
Okay, this may be an eye opener, but thanks to my pedagogical failure here, you need to put glasses on! There is definitely potential for confusion here, and I’ve been struggling to figure out when to address it (and maybe I need to address it this week.)
The possibility space is not (x1, p1) space — that’s called “phase space”. The space of possibilities I’m referring to is sometimes called “configuration space”, just x1. See this article for details. For one particle on a line, the physical space is 1d, and so is the possibility space.
The distinction between physical space and the space of possibilities (i.e. configuration space) becomes obvious when you have N particles on a line, because the physical space is still 1d, but the space of possibilities is N-dimensional (and the phase space is 2N dimensional.)
Wave functions exist in the space of possibilities/configuration space, not in phase space. You can also write a wave function in the “dual” or “conjugate” space of possibilities, where instead of writing it as a function of x1 (with no reference to p1), you write it in terms of p1 (with no reference to x1). This is called momentum space. So you have a choice: you can write the wave function in configuration space, or in momentum space, but not in phase space.
If you write the wave function in configuration space, the wave function’s shape tells you the distribution of possible measurements of x1, and the derivative of the wave function d Psi(x1)/dx1 tells you the distribution of possible measurements of p1. (If you use momentum space, the reverse is true.)
The only thing that is conserved for a wave function is probability; once normalized so that the integral of its square over the space of possibilities is equal to 1, that will remain true at all times, as long as QM is valid for the system. Uncertainties are not conserved, so there is no need for a broader x1 distribution to lead to a narrower p1 distribution. You can see that in my examples in this post, especially the last one.
Hopefully now I’ve given you the right glasses, and your eyes are not only open but also properly focused too!
There’s nothing wrong with “obsessing” about making things clearer.
Seán
These articles are so good and really help me to better appreciate your book. I especially liked the article you linked to above, which addressed ( to a limited degree ) two things I’ve been thinking about. How to visualize two wavicles and how to visualise the wave function in QFT.
I have always felt far too dumb to understand QFT. I think really because I find it too hard to visualize. The books I’ve read just seem to reduce to a manipulation of symbols which leaves me feeling dissatisfied.
I notice you go from QM to QFT and skip over RQM. Is this because RQM is a confusing distraction ?
Should I visualize the wave function in QFT as being static ( no time dependance ) ?
Relativistic quantum mechanics isn’t consistent; relativity needs you to be able to make particle pairs, but you can’t in QM, relativistic or not. This approach was good as an intermediate step, but there’s no consistent theory there. You need QFT. The reasons are discussed at the end of this post.
Wave functions in QFT are time dependent and satisfy a Schroedinger equation also. We’ll see that soon enough.
However, just as in QM, one can choose the “Heisenberg picture” in which wave functions are static and operators change with time, as opposed to Schroedinger’s picture where the reverse is true.
But if you want to preserve special relativity at every step, you will want to use Feynman’s innovative approach — the approach of path integrals. I haven’t decided if and when to discuss that, but it’s many months off.
Fascinating. One of the aspects of QFT that I’ve found frustrating in the text books, is the lack of any clear description of what a “field” actually is and how it relates to the wavefunctions of QM. This is precisely the kind of introductory chapter that the textbooks need.
David Tong does actually show how to construct the QM’s wavefunction from QFT, but it’s very much an aside from the main body of his lectures. Most other texts don’t even consider it.
Very much looking forward to your next post. After that, I’m hoping I’ll have some sensible questions.