# Does QFT re-interpret the meaning of the wave function of Schrodinger's equation?

I'm wondering if quantum field theory re-interprets the meaning of the wave function of Schrodinger's equation. But more specifically, I'm trying to understand how to explain the double slit experiment using quantum field theory's interpretation that, in the universe, "there are only fields."

As background, in this post, Rodney Brooks states:

In QFT as I learned it from Julian Schwinger, there are no particles, so there is no duality. There are only fields - and “waves” are just oscillations in those fields. The particle-like behavior happens when a field quantum collapses into an absorbing atom, just as a particle would.

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And so Schrödinger’s famous equation came to be taken not as an equation for field intensity, as Schrödinger would have liked, but as an equation that gives the probability of finding a particle at a particular location. So there it was: wave-particle duality.

Sean Carroll makes similar statements, that the question "what is matter--a wave or a particle?" has a definite answer: waves in quantum fields. (This can be found in his lectures on the Higgs Boson.)

In the bolded passage above, Dr. Brooks seems to suggest that QFT provides a physical interpretation which removes superposition. And he says as much in another post here:

In QFT there are no superpositions. The state of a system is specified by giving the field strength at every point – or to be more precise, by the field strength of every quantum. This may be a complex picture, but it is a picture, not a superposition.

So taking up the double slit experiment, is the following description accurate? When the electron passes through the double slit, waves in the electron quantum field interfere. When the wave collapses into a particle, it takes on the position at one of the locations where the electron quantum field is elevated. So the electron particle can't "materialize" in any locations where the electron quantum field interferes destructively. This gives rise to the interference pattern on the back screen.

Is this a correct description of the double slit experiment from QFT's interpretation that, in the universe, "there are only fields"? If this is correct, then it seems like QFT says the wave function is more than just a probability wave: the wave function describes a physical entity (excitations in the underlying quantum field). There is still a probabilistic element: the position where the wave collapses into a particle has some random nature. Am I understanding correctly that QFT adds a new physical entity (quantum fields) which expands our physical interpretation of the wave function?

• duplicate of physics.stackexchange.com/q/343237/84967 Commented Nov 5, 2017 at 16:27
• @AccidentalFourierTransform, Q343237 seems to ask if the ripple = the wave nature and its localization = the particle nature. It's a question about wave-particle duality. I believe my question is more about the physical interpretation of a quantum field as it relates to the wave function. I don't intend to ask about wave-particle duality specifically. I'm sure you can read the answers in the linked question and see how they relate to my question, but I do think there are extra steps connecting the two which I don't see, and which may be enough to qualify this as a separate question. Commented Nov 5, 2017 at 16:42
• @AccidentalFourierTransform, further to the distinction between my Q and the linked Q, the interpretation I've cited is that there are no particles--just waves. Again, there's probably a way to use the cited Q and its answers to address the Q I've asked, but I wonder if the differences are enough to qualify as a distinct question. Commented Nov 5, 2017 at 16:44
• See the answer by Motl here physics.stackexchange.com/q/4212 . BTW there is no elevations of the electron quantum field, just creation operators acting on the quantum field create an electron at (x,y,z.t) Commented Nov 5, 2017 at 17:30
• @annav, thanks! I'll check this out. So when people like Sean Carroll say "particles are excitations in a quantum field," are they speaking metaphorically, and not describing a physical picture? And is Brooks incorrect, then, in suggesting to interpret it as an "equation for field intensity"? Commented Nov 5, 2017 at 17:51

There is an overlap with other questions linked in the comments. But, perhaps the focus of this question is different enough to merit a separate answer. There are at least two distinct but equivalent formalisms of QFT: the canonical approach and the path integral approach. Although they are equivalent mathematically and in their experimental predictions, they provide very different ways of thinking about QFT phenomena. The one most suited to your question is the path integral approach.

In the path integral approach, to describe an experiment, we start with the field in one configuration and then work out the amplitude for the field to evolve to another definite configuration representing a possible measurement in the experiment. So in the two-slit case, we can start with a plane wave in front of the two slits representing the experiment starting with an electron of a particular momentum. Then, our final configuration will be a delta function at the screen representing the electron measured at that point at some later specified time. We can determine the probability for this to occur by evaluating the amplitude for the field to evolve between the initial and final configuration in all possible ways. We then sum these amplitudes and take the absolute value squared in the usual QM way.

So, in this approach, there are no particles, just excitations in the field.

• This is extremely helpful and easy for someone with some (but limited) background like me to follow. One follow-up question: is it fair to say that the field is a physical entity? Or would this qualify as an interpretation which goes beyond the math? Perhaps it's just the wrong question to ask, because the math is agnostic about an answer. Your insights into how the path integral fans approach that question would be much appreciated. Commented Nov 5, 2017 at 18:23
• From a path integral perspective, the field is a physical entity. We then apply QM to it to evaluate the probability of it evolving from an initial to a final state. Commented Nov 6, 2017 at 0:21
• So it seems that, under this interpretation, my description above is incorrect. We can't use interference as the mechanism for explaining why the field evolves the way that it does. We still have superposition of many possible outcomes, and these outcomes describe many possible evolutions of the field which collectively produce an interference pattern. If you could confirm that I've got this right, I'll go ahead and accept your answer. Commented Nov 6, 2017 at 0:28
• I think your explanation of interference is the standard QM one. From a QFT perspective, it is not quite right. But for slow-moving electrons, QM should give a good description and so it is approximately right mathematically. But, if the electrons are fast moving you would need to work it out with QFT. Commented Nov 6, 2017 at 0:46
• I think that is approximately correct in the low-velocity limit. Commented Nov 6, 2017 at 1:37

The basic division in the physics models used to describe nature came with quantum mechanics and its postulates. The divide is the deterministic predictions of the classical theories, mechanics, electrodynamics, and the probabilistic predictions of quantum mechanics which postulate that there exists no underlying level of determinism.

This means that what is considered a particle of mass m in classical physics has a specific (x,y,z,t), whereas in situations where quantum mechanics is necessary there is only the probability of finding a particle at (x,y,z,t) given by the solution of the quantum mechanical differential equation and its boundary conditions.

The Schrodinger equation is very successful in describing the underlying quantum mechanical probabilistic dynamics, there exists mathematical continuity, the classical level emerges from the underlying quantum mechanical, similar(hand waving) to how thermodynamics emerges from the mathematical model of statistical mechanics.

Then special relativity enters the mathematical framework of quantum mechanics and new equations and mathematical methods are necessary to be able to calculate and give predictions for relativistic energy experiments.

What is a field in physics?

In physics, a field is a physical quantity, represented by a number or tensor, that has a value for each point in space and time

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Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other independent physical variables on which the field depends. Usually this is done by writing a Lagrangian or a Hamiltonian of the field, and treating it as the classical mechanics (or quantum mechanics) of a system with an infinite number of degrees of freedom. The resulting field theories are referred to as classical or quantum field theories

Quantum field theory is a mathematical tool that is used on an set of underlying complete solutions of a quantum mechanical equation,( not only for particle physics,) to calculate interaction probabilities at relativistic energies.

It uses the standard model Lagrangian for point particles to define a field for each particle covering the whole of space time, an electron field, a neutrino field etc. On these fields particle creation and annihilation operators work , so that an electron or muon or photon ... is observed with a calculable probability. These fields are the plane wave solutions of the quantum mechanical equation for the particles. On these, creation and annihilation operators generate a propagation of the particle in space, and particles are considered as an excitation of the field.

Feynman diagrams are the calculational tool describing interactions of particles and predicting crossections and decay rates very successfully using the standard model.

It is a mathematical picture. An creation operator on the electron field generates an electron at a specific point. To get the heisenberg uncertainty of a real particle one has to use wave packet formulations . As stated in this answer , there is continuity between the relativistic frame and the low energy frame where the double slit experiments can be studied.

For the non relativistic case, the double slit is the wavefunction solution of boundary conditions : "electron scattering on two slits with specific dimensions"

You ask in a comment to your question:

So the new mathematical picture of QFT fails to imply any new physical interpretation beyond what QM already provided?

In my opinion yes, the basic divide in interpretation are the postulates of quantum mechanics , the probabilistic versus the deterministic predictions of mathematical models to fit physics.

To posit that particles are just excitations of a field, on which creation and annihilation variables work, and the measurable partilces wavepackets (which still are under research) on a field, is a useful model as long as it predicts physical interactions. To set particle fields as the underlying true level of nature becomes metaphysics, a belief ; in addition it introduces a new Lorentz covariant ether, and we know the fate of the classical ether. I prefer to think of QFT as a useful mathematical tool.

The statement " the probabilistic versus the deterministic predictions of mathematical models to fit physics" are a basic divide common in all quantum mechanical calculations and refer to the models fitting the data.

"Interpretations" are a meta level, based on beliefs and preferences and not on models fitting data, they belong to metaphysics.The only reality are the numbers to be fitted and predicted, imo.To get a taste of what the future may hold for models fitting data have a look at this . It is still quantum mechanics.

• I've been thinking about your argument that interpretations of the math belong to metaphysics, not physics. Do you think this is true for other mathematical constructs that physics uses--like, perhaps, energy? Is it wrong to say that energy is anything more than just a mathematical quantity, just as quantum fields are simply mathematical quantities? Is energy similarly devoid of any true underlying level of nature? I'm really curious if you think there's a meaningful distinction between energy and quantum fields, and if so, what that difference is. Commented Feb 24, 2020 at 5:35
• Energy is defined first in the classical frame and is used as one of the coordinates in the energy-momentum four vector, still a classical quantity. Like x, y,z,t it is defined by measurement and is axiomatic . Quantum mechanics used the same coordinates to develop probabilistic models still using the same variables for its coordinate systems, because all experimental results are expressed where we live , the classical frame. Commented Feb 24, 2020 at 5:50