# Wave/particle-duality as result of taking different limits of a QFT

There is an account on dualities in quantum field theories and string theories by Polchinski from last week

http://arxiv.org/abs/1412.5704

At the end of page 4, he writes the wave/particle dichotomy arises from different limits you can take in a quantum field theory.

Which limits are meant here exactly, and can one give a proper example? I assume it might relate to many/few quanta states.

• The particle picture arises from QFT e.g. by looking at the path that gives the larges contribution to the action in a Feynman path integral, which happens to be the classical path. Experimentally one can arrive at this with weak measurements e.g. in a cloud or bubble chamber on a single particle, with each interaction changing the momentum of a particle very little. This works on a "single" high energy particle (although there are still a lot of individual quantum processes!). The wave picture emerges by looking at the collective of many quanta, each of which makes a single interaction. – CuriousOne Dec 22 '14 at 15:06
• I would be interested in seeing a for the interested amateur type answer to this. That is, an answer that expects you to work at it but doesn't assume any specific knowledge of QFT. If the consensus is that such an answer is possible/useful I'd be willing to place a bounty on the question. – John Rennie Dec 23 '14 at 10:54
• @CuriousOne : The least action principle is known from classical mechanics. But I have doubts that picking a path gives a "particle picture". One can pick whatever one wishes. With classical bodies we can deprive the object from the other paths, i.e. limit its evolution to that one path, and it's O.K. But with a quantum object, if we only dare to limit its evolution at one point of that path, to $\Delta r = 0$ the particle subsequently may follow whatever paths in the universe. – Sofia Dec 23 '14 at 11:19
• OP here. The paper suggestion you have two limits, a particle limit and another field limit. I'd like to know/see both different limits, and explicitly with some QFT. – Nikolaj-K Dec 23 '14 at 13:05
• I've placed a (large!) bounty on this because I see it as an important contribution to writing the definitive article on wave particle duality. An answer targeted at the mathematically sophisticated amateur (like me :-) would be ideal. An answer of this type is likely to be long, because I'm guessing lots of side issues will also need to be explained. But then I'm offering the maximum bounty, and you have the Christmas/New Year holiday to write it in :-) – John Rennie Dec 26 '14 at 7:31

There are probably various answers to this question and I will try to provide one that I consider quite interesting. It is a specific realization/example of the fact that the path integral is dominated by estrema of the action.

The wave aspect of a QFT is probably trivial as QFT is dealing with wave equations. This is particularly apparent for massless particles and I will not discuss it any further.
Let me thus focus instead on the opposite limit when the particles are very heavy. I will use the Schwinger proper time and heavily follow Matt Schwartz textbook.

For simplicity, consider the propagator of a scalar particle in an external field source $A_\mu$ that in the Schwinger proper time takes a path-integral form over the particle trajectory $$G_A(x,y)=\langle A|T\phi(x)\phi(y)|A\rangle=\int_0^\infty ds e^{-is m^2}\langle y| e^{-i\hat{H}s}|x\rangle$$ where $$\langle y| e^{-i\hat{H}s}|x\rangle =\int_{z(0)=x}^{z(s)=y} [dz(\tau)] e^{i\mathcal{L}(z,\dot{z})}$$ with $$\mathcal{L}=-\int_0^s d\tau \left(\frac{dz^\mu(\tau)}{2d\tau}\right)^2+e \int A_\mu(z) dz^\mu\,.$$ It is convenient to rescale the variables with the mass, $s\rightarrow s/m^2$ and $\tau\rightarrow/m^2$ so that the path-integral is clearly dominated by the free kinetic energy when the mass is large $$G_A(x,y)=\frac{1}{m^2}\int_0^\infty ds e^{-is}\int_{z(0)=x}^{z(s/m^2)=x} [dz(\tau)]e^{-i\int_0^s d\tau m^2(\frac{dz^\mu}{d\tau})^2+i\int eA_\mu dz^\mu}$$ This is the limit of particle that takes a well definite trajectory since the path-integral is dominated by the point of stationary phase that corresponds to the free particle solution $$z^\mu(\tau)=x^\mu+\tau v^\mu\qquad v^\mu=(y-x)^\mu/s\,.$$ Moreover, on this solution the propagator becomes (after rescaling back to the original variables) $$G_A(x,y)=\int_0^\infty ds e^{-i\left[s m^2+\frac{(y-x)^2}{4s}-ev^\mu\int_0^s d\tau A_\mu z(\tau)\right]}$$ where the last term is the same that one get by adding the source current $$J_\mu=v_\mu \delta(x-v\tau)$$ so that the heavy particle creates the field $A_\mu$ as if moving in a classical trajectory at constant speed. As Schwartz says, when a particle is heavy the QFT can be approximated by treating the particle as a classical source (but treating everything else as quantum, e.g. the particle can possibly generates quantum radiation $A_\mu$ upon which we haven't integrated over yet).

• tl,dr: The stationary phase approximation is better for heavier, i.e. more classical, particles. Therefore you can understand the physics in that limit by considering only the extrema of the action, which is like tracing the path of a particle. – DanielSank Dec 29 '14 at 16:58
• @DanielSank I think you don't realize that extrema of the action can describe a priori also classical wave solutions, not necessarily particle-like solutions. In my answer I show a particular limit where the extremum of the action of a (quantum) field theory gives in fact a (classical) particle-like behavior as opposed to a (classical) wave behavior. – TwoBs Jan 1 '15 at 17:41
• Good point! I did not appreciate that. Thanks. – DanielSank Jan 1 '15 at 17:42

Wave particle duality is not a quantum physical issue! Here is a full description of its simple mechanism, exclusively based on special relativity which is easily understandable by any interested person.

Wave-particle duality is deeply embedded into the foundations of quantum mechanics (Wikipedia).

This statement is entirely disproved in the following by showing one case which may entirely be explained classically: light in vacuum.

The following derivation is based exclusively on the two postulates of special relativity from which is resulting directly and compellingly the entire model for light in vacuum.

There is one unexplored zone in special relativity which seems to yield only meaningless results. When particles are moving not only near speed of light (v < c) but at speed of light (v=c), the Lorentz transforms cease to operate. The proper time is reduced mathematically to zero, but there is no reference system from which this could be observed. Also, lengths would be reduced to zero for such a hypothetical non-existent reference system.

As a consequence, up to now the corresponding equations deriving from special relativity (time dilation and length contraction) were simply confined to massive particles, excluding the case v=c from the domain of definition of these equations. There is no physical legitimation for such a break in their application (implying de facto a limitation of universal validity of special relativity), and Einstein's special relativity does not cease to exist at v=c as it is shown by the means of an example in the following chart:

By consequence, it follows from the equations for proper time and length contraction that a photon which is traveling the distance Sun-Earth according to our observations in t=8 minutes for a distance of s=8 light minutes, has from its (hypothetical) own point of view a proper time t'=0 and travels a distance s'= 0.

If time and traveled distance are both zero that would mean that there was no movement. When I am traveling zero meters in zero seconds, I did not move, and there is no movement which could be subject to a measurement of velocity. My velocity is not defined (0m/ 0 sec.)

The Lorentz factor splits realities

The twin paradox shows with unequaled clarity the effects of the Lorentz factor.

Example: A twin brother undertakes a space travel and returns after 20 years. At his return to Earth the twin brother who remained at home observes that the traveling twin aged only by 5 years.

In this example the observed time on the observer's clock is 20 years. The proper time (and thus the real aging) is only 5 years instead of 20 years. These two realities are linked arithmetically by the proper time equation and by the Lorentz factor.

Moreover we can notice a hierarchical order of realities: We cannot say that the traveling twin has become 20 years older, even if all observers on Earth have measured 20 years. This would be in contradiction with the physical condition of the traveling twin who looks younger than the twin who stayed on Earth. This means with regard to photons that the proper reality of the photon, even if it may not be observed by anyone, reflects its primary reality. All observations are secondary with regard to this primary reality. Even the constant of speed of light c.

By consequence, and in accordance with the wording of the second postulate of special relativity, light velocity c is a secondary observer's reality. We observe a movement of light which according to the primary reality of the photon is a standstill.

The Lorentz factor is assigning to photons two realities, that means, the transmission of the light momentum is double-tracked:

The secondary reality is the (commonly known) observed reality: Maxwell equations are describing a light quantum in the form of an electromagnetic wave moving at speed of light (v = c, t = 8 min, s = 8 light minutes). The transmission of the momentum occurs indirectly from Sun to the wave and then from the wave to Earth.

The primary reality is the unobserved proper reality of the photon: t'=0 and s'=0, proper time and distance are zero, there is no velocity. That means that the momentum is transmitted directly outside of spacetime from Sun to Earth, without intermediate medium.

Result:

1. A classical explanation of Young's double slit: while we are observing nothing but an interfering wave, the particle characteristics of light in vacuum are transmitted directly (path length = 0) and in parallel to the electromagnetic wave.

2. Light in vacuum is a primitive border case of quantum physics which can be explained classically. As a result, the mere wave-particle duality can be described without non-locality issue (see also the open (former bounty) question) as a classical phenomenon.

3. This fact does not change at all quantum physics with all its non locality issues. But it shows that there is one classical case of wave-particle duality, with no need of recourse to quantum mechanics and/ or QFT.

4. A simple answer to the question of NikolajK and John Rennie what the nature of wave-particle duality is.

• I appreciate the effort you've put in, but your answer seems to be unrelated to the original question or to the targets I laid out for the bounty. Just to clarify, while I'm interested in the question because I have a deeper interest in wave-particle duality, to earn this bounty you need to answer Nikolaj's question. – John Rennie Dec 26 '14 at 16:11
• @john rennie: No problem, I understand! Anyway thank you for this very nice Christmas bounty you offered to Stack Exchange users, I find this a very good idea! For your private interests in wave-particle duality, I remain at your disposal with regard to my text. – Moonraker Dec 26 '14 at 16:26
• What you call different "realities" are just different coordinate systems, it's fundamentally no different from the fact that you can describe the same Newtonian scenario with different Galilean coordinate systems that assign different x and y coordinates to a given event. And I don't see how your answer gives a non-quantum version of wave-particle duality, since classical electromagnetic waves aren't measured to set off detectors at highly localized positions like individual quanta (photons, electrons) are. – Hypnosifl Dec 31 '14 at 20:28
• @hypnosifl : Within the first reality there is no coordinate system, instead there is a banal pointlike reality which, however, is real. - Wave-particle duality of photons in vacuum can be explained mathematically and by analogy with the classical twin phenomenon ---- not to be confounded with the fact that photons may be subject to measurements of quantum physics. Photons in vacuum may be considered as primitive border case of quantum physics, characterized by their empty space time interval. – Moonraker Jan 1 '15 at 15:56
• "Wave-particle duality of photons in vacuum can be explained mathematically and by analogy with the classical twin phenomenon" -- Your answer doesn't make clear what precisely this analogy is supposed to consist of, it all seems rather handwavey. What exactly is the SR analogue of the "particle" aspect of the photon and what exactly is the SR analogue of the "wave" aspect, and what is the SR analogue of the "duality" aspect where a given experimental setup will only reveal one or the other? – Hypnosifl Jan 1 '15 at 16:55

These might be two different issues. Wave-particle duality is one issue, different classical limits is another issue.

Wave-particle duality often refers to the fact that when choosing an experiment historically, people sometimes chose options that revealed wave properties and sometimes chose options that revealed particle properties. So the hypothesis was that nature has both qualities waiting to be revealed by different choices of experimental setups that measure the same initial input.

As for classical limits, (assuming you aren't doing MIW or dBB) a classical limit is one where you can ignore (relative) phases (classical fields and particles are entirely real, they have no phase).

For a bosonic field, you can take something like a classical wave limit. You have the option to take a high quantum number limit that is also a coherent state, then there is no relative phase, so the phase can be ignored, and it looks like a classical field. So it's not just a high quanta limit, you also need the coherence. I didn't go into much detail because Motl seems to cover it in detail at the level you are looking for in http://motls.blogspot.com/2011/11/how-classical-fields-particles-emerge.html

You can also take something like a classical particle limit, this is a low quanta limit but also a limit where the energy is kept high. So for the electromagnetic case, this would be single gamma rays, and now the scattering of a single quanta (where QFT reduces to just relativistic quantum mechanics since there is only a single quanta). In this limit the phase doesn't matter for the scattering angle, and you can compute it as compton scattering by a photon of fixed momentum $h\nu$. The details about the QFT to RQM limit (single quanta) is well known and how the high energy RQM scattering reduces to that of compton scattering I think is simply because there are so few options that conserve energy and momentum and scattering states have to be on shell. Again, probably well known.

None of this is as deep as I think you expected, but I wanted to provide the filling in of what I thought the authors meant and it might be things you probably already knew but they just didn't give enough details for you to know it was stuff you already knew.