1. A Brownian motion particle is a random particle, whose (probability density of) location can be described by heat equations. This is not surprising to me, since heat equation basically captures some "stuff" that propagate but with the sum of certain attribute being conserved. In the case of heat propagation, it is the heat that is being spread and conserved; whereas in the case of Brownian motion particles, it is the (probability density of) location of the particle to be spread and conserved.

  2. The free theory of $1$-dimensional QFT (seems to can) be described by heat equation too.

Is it possible that particles are just particles, which are not too far from what we have been imagined before QM has developed? And they appear to be like waves simply because they are "random"?

When interactions have to be considered, they are not "freely random" anymore, thus no longer can be described by a clean heat equation? However, its random nature is still there, thus its "wave-like" nature?

While lots of tiny particles "have to" stick together (required by suitable interaction terms), though still random, the randomness of the whole entity decreases, giving us the classical theories?

Difficulties of my (non-sense) guesses above

  1. $1$-D free QFT has a rigorously defined measure (Wigner). Probably my picture in mind does not even make sense for higher dimensional theories.

  2. I don't know how to make my picture in mind rigorous for non-free theory.

  • 1
    $\begingroup$ Let's start with the following: "The free theory of 1-dimensional QFT (seems to can) be described by heat equation". What specifically do you mean when you say this? $\endgroup$ Commented Nov 19, 2019 at 14:30
  • $\begingroup$ I should have been more clear on this: I meant the fact that the correlation function $P(x,y)$ is the integration in $t$ of the heat kernel $K_t(x,y)$. A reference is Costello's renormalization and effective field theory chapter 2.6. $\endgroup$
    – Student
    Commented Nov 19, 2019 at 14:37
  • $\begingroup$ "Freely random" does not explain the regular, wave-like interference patterns that appear in those experiments and instruments that make use of and/or reveal them. $\endgroup$ Commented Nov 19, 2019 at 14:50
  • $\begingroup$ @SolomonSlow right, because there are interaction terms in interference experiments. $\endgroup$
    – Student
    Commented Nov 19, 2019 at 14:53
  • 2
    $\begingroup$ @Student The heat equation indeed has some similarities with the Schrödinger equation (SE). However, the coefficient of the time-dependent part contains an imaginary $i$. This makes a big difference. $\endgroup$ Commented Nov 19, 2019 at 16:56

2 Answers 2


The fact that a beam of photons or electrons can produce an interference pattern with dimensions measured in centimeters, leaves no doubt that they have wave properties, and if you want to read a book on quantum mechanics you need to be comfortable with wave equations expressed as imaginary functions. The particle idea is associated with the fact that all of the energy and momentum of a “wave packet” can be absorbed by an entity the size of an atom or smaller.



It is possible to make sense of the wave-like behavior in a pure particle context.

In this paper:

The Schrodinger Equation, the Zero-Point Electromagnetic Radiation, and the Photoelectric Effect H. M. Franc, A. Kamimura, G. A. Barreto Braz J Phys (2016) 46:184–191 DOI 10.1007/s13538-016-0398-3

we read (page 190):

"Notice that the equations (36) and (37), and the superposition principle suggest that the interference phenomena of electrons, and other matter particles, may be explained without the use of de Broglie waves"

Another paper:

Tunneling as a Classical Escape Rate Induced by the Vacuum Zero-point Radiation A. J. FARIA, H. M. FRANC and R. C. SPONCHIADO  Foundations of Physics 36(2) · October 2004


"the classical escape rate calculation presented in our paper gives results entirely analogous to the quantum tunneling description, provided that the electromagnetic zeropoint fluctuations are included in the calculations."

So, again, we see that a description in terms of wave-like behavior (tunneling) can accommodate a particle description (with the necessary presence of external E-M fields).


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