Anti-matter as matter going backwards in time? (requesting further clarification upon a previous post)

Question

I understand this question has already been asked here, however, I don't have enough reputation points to place a comment (I suppose that's the reason) on a specific answer to request a reference.

A user has answered (I'm not sure I can cite his name here, so I won't):

There is one technical inaccuracy in saying that antimatter moves back in time (whatever it might mean). In quantum field theory we get positive energy solutions (usual particles) and negative energy solutions. Negative energy solutions behave in time as if they were propagating backward in time. But they are not the antiparticles, they are just the "negative-energy particles". Antiparticles are positive energy solutions, and they are obtained by acting with charge conjugation operator on the negative-energy solutions. So, antiparticles move forward in time, as usual particles.

and, even though someone else comments

This is interpreting "going back in time" differently than anyone else does.

I'd like a reference for (or further clarification on) that, since I have already heard that before (also informally).

I understand this is a repost but if someone could at least give me some reference before closing it, I'd be thankful.

PS: My first intention was to ask how to proceed on this case in "meta", but I don't have reputation points to place a question there.

EDIT:

To clarify further my question: I had originally the same question as the OP from the post I quoted. At first, I thought that an explanation of what are anti-particles and the interpretation as particles going back in time could be found in quantum mechanics alone, without more advanced quantum field theory. The argument I knew is that anti-particles are really related to the originally negative-energy solutions, that can be reinterpreted as positive-energy solutions, as long as one invert the sign of time in the unitary time-evolution operator. This would hence lead to the reinterpretation of such states as positive-energy anti-particles going back in time or, at least, with anomalous time dependency.

The particular answer I quoted, seems to contradict this interpretation, saying that it is a mistake. I'd like a further clarification here. Is this interpretation really wrong?

From what is said on Wikipedia, looks like the Feynman–Stueckelberg interpretation relies on the fact that anti-particles are to be understood as particles going backwards in time and then related to the latter by charge conjugation.

Where does this interpretation primordially comes from? Where is it primordially needed if their association with the negative energy solutions of Dirac's equation (say) is a mistake? I understand that in quantum field theory one uses complex fields to describe both particles and anti-particles, relating them by charge conjugation. But isn't this done this way to accommodate Feynman–Stueckelberg's interpretation?

Answer 1

Please bear in mind that I am an experimentalist, therefore I treat theoretical ideas and models as dependent on experimental observations and not vice versa.

Facts:

• elementary particles have measurable quantum numbers. These quantum numbers define the particle.

• elementary particles have mass

• there exist elementary and composite (from elementary) particles that have the same mass as others but opposite charge, as the electron vs positron, proton vs antiproton, called antiparticles because:

• there is a very high probability that when scattered against each other, the particle and antiparticle disappear and the energy appears in a plethora of other particles. This is called annihilation because charge disappears, and in general the opposite quantum numbers are "annihilated".

On the theoretical side these experimental observations are fitted beautifully by the Dirac equation if one considers the negative energy solutions to describe the antiparticles.

One thing we can be sure of is that measured antiparticles travel forward in time.

The popular theoretical interpretation of antiparticles being particles traveling backwards in time mainly comes from the Feynman diagrams. These are a brilliant mathematical tool for fitting and predicting measurements representing particle interactions as incoming lines and outgoing lines and in between lines representing virtual particles that carry the quantum numbers but are off mass shell.

Due to the CPT theorem, once a Feynman diagram is drawn, one can interpret the lines consistently according to CPT and will get the corresponding cross sections and probabilities for the change in the quantum numbers consistent with CPT (charge conjugation, parity and time reversal). Identifying a positron as a backwards-in-time electron is an elegant interpretation that in the Feynman diagrams exhibits the CPT symmetry they must obey.

What I am saying is, the statement "positrons are backward-going electrons" is a convenient and accurate mathematical representation for calculation purposes. "As if". There has not been any indication, not even a tiny one, that in nature (as we study it experimentally) anything goes backwards in time, as we define time in the laboratory.

Edit replying to comment by Nathaniel:

I'm curious: how would you expect empirical data from a backwards-travelling positron to differ from what we actually see?

In this bubble chamber picture we see the opposite to annihilation, the creation by a photon of an e+ e- pair. (This is an enlarged detail from the bubble chamber photo in the archives. The original web page with the letterings has disappeared, as of july2017)

The magnetic field that makes them go into helices is perpendicular to the plane of the photo. We identify the electron by the sign of its curvature as it leaves the vertex. The positron is the one going up to the left corner. We know it is not an electron that started its life before the vertex formed because as an electron/positron moves through the liquid it loses energy and the loss defines the time direction of the path. So the particle has to start at the vertex and end at the upper left, so it has the opposite curvature to the electron and it is a positron.

A Feynman diagram looks like a scattering in real space, or a pair production, but one cannot project the intricacies of the mathematics it represents onto real space. It is only the calculations of cross sections and probabilities that can be compared with measurements.

Answer 2

Yes, antimatter particle can be treated as moving back in time in some sense.

But consider some points:

(1) If an abject is moving "back" in time, we, normal objects, will see it as moving forward in time. For example, if some person lives back in time, we will see him or her as living forward in time, but being first dead, then old, then adult, then child and finally born. I.e. we will see a person who has reverse order of life stages. Suppose the Earth is full of persons which live forward or backward. What we will decide? We will decide that there are two types of persons with reverse order of stages, nothing more.

(2) Particles have no life stages like persons. Both electron and positron do not change or evolve while "living". They are unchanged and have constant properties. Also, the process of "moving" (in space) does not have time direction.

So, reversion of anti-particles just means that they have some 4D (space-time) properties, and these properties are oriented opposite for anti-particles.

(3) Particle physics probably obeys so called CPT symmetry. I.e. each possible "chemical" reaction between particles, can be (P) reflected in a mirror, (T) reversed in time and (C) particles replaced by antiparticles and this procedure will give also possible "chemical" reaction.

So, if you say that antiparticles are particles reversed in time, you are saying that they obey CT symmetry, which is not fully true. Full symmetry is CPT, so antiparticles are reversed in time and reflected in a mirror.

Answer 3

The notion of anti-particle emerges as soon as special relativity is taken into account. For a relativistic particle of mass $m$, its energy and momentum satisfy $E^2 = p^2 + m^2$. To illustrate where the anti-particles are coming from, let's take the simplest "wave function equation" (that's an abuse of language since it's an equation for fields), i.e. the Klein-Gordon one. For simplicity I consider only 1 dimension: $$\frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} + m^2 \phi = 0$$ A solution of this equation is: $$\phi_1 = e^{-i(Et-p.x)}$$ since we have $\frac{\partial^2 \phi_1}{\partial t^2}= (-iE)^2 \phi_1 = -E^2 \phi_1$ and $\frac{\partial^2 \phi_1}{\partial x^2} = (-ip)^2 \phi_1 = -p^2 \phi_1$ and $E^2 = p^2 + m^2$. So far, so good. Now, there is another solution of the K.G. equation given by: $$\phi_2 = e^{+i(Et-p.x)}$$ with still $E^2 = p^2 + m^2$. Thus, 2 solutions describing 2 particles with same mass. What the difference between these 2 particles? The charge. It can be shown that the charge is given by this expression: $$q(\phi) = i(\phi^* \frac{\partial \phi}{\partial t} - \phi \frac{\partial \phi^*}{\partial t})$$ By injecting $\phi_1$ and $\phi_2$ you find: $$q(\phi_1) = +2E,~~~~q(\phi_2) = -2E$$ You might be surprise that the charge depends on the energy but what matters here is that the 2 charges are opposite (the fact that it is proportional to $E$ is because of relativistic normalization). Conclusion, $\phi_2$ seems to describe a particle having all the property of the one described by $\phi_1$ except that its charge is opposite: it is the antiparticle of $\phi_1$. Now, notice that I can express $\phi_2$ with $\phi_1$. Indeed, I just have to reverse the sign of $E$ and $p$: $$\phi_2(E,p) = \phi_1(-E,-p)$$ So if $\phi_1(E,p)$ describes a particle with a positive energy $E$ and momentum $p$, the antiparticle can be also described by $\phi_1$ with a negative energy and an opposite direction of the momentum. But: $$\phi_1(-E,-p) = e^{-i(E(-t)-(-p).x)}$$ and thus the antiparticle expressed with $\phi_1$ seems to go backward in time $(-t)$ and with the momentum $-p$. That's all! An anti-particle can be described by the solution of a particle if you reverse the time (or the energy) and the momentum. Nothing more. But obviously, everybody prefer to describe the antiparticle with $\phi_2$ with meaningful physical quantities: the antiparticle travels forward in time (as any particles).