What I learned as a hired consultant to autodidact physicists (2016)
Thanks Professor Hossenfelder.<p>You mentioned that the amateur physicists don&#x27;t know enough math. Well, some of us amateur physicists have concluded that the academic physicists don&#x27;t know enough math.<p>E.g., when I was trying to be a ugrad physics major, the prof in the E&amp;M course tried to prove the divergence theorem.  His proof was, in one word, junk.  Sadly as a ungrad math major, the math department didn&#x27;t prove it either.  By now, sure, I&#x27;ve gone through fairly careful treatments including the Jacobian, ..., exterior algebra of differential forms (texts by Fleming, Rudin, Apostol, ...).  Even some of those treatments are a bit fishy in places and incomplete:  Physicists keep using old versions of <i>differentials</i>, and the math books got rid of those 50+ years ago; clean ups are not too difficult but need to be done by either physics or math.<p>The Riemann integral over the whole real line is fishy (smells like old fish).  The <i>improper</i> integral approach of integrating from <i>-a</i> to <i>a</i> and then taking the limit as <i>a</i> goes to infinity is also fishy.  There is a good clean up -- the Lebesgue theory.<p>Back to the divergence theorem, physics wants to have a 2 dimensional measure they write as <i>dA</i> and a three dimensional measure they write as <i>dV</i>, but there is a gap in their argument:  They are necessarily working with what measure theory calls <i>product measure</i> and in particular Fubini&#x27;s theorem, and there needs to be some orthogonality that is not made clear, in either math or physics.<p>I&#x27;m tired of screaming myself to a sore throat over the claims that the wave functions of quantum mechanics form a Hilbert space.  Hilbert space is a math idea, and in math departments we know quite well what a Hilbert space is, a complete inner product space.  The &quot;complete&quot; part is in the sense of Cauchy, that every Cauchy convergent sequence converges.  Well, the wave functions fail that completeness assumption.<p>You mentioned that you can tell backgrounds just from use of language:  Right, so can math people!  So, when an MIT physics guy teaches quantum mechanics and says that &quot;the wave functions are differentiable and also continuous&quot; we know his background is in physics, not math -- EVERY differentiable function is continuous.<p>It goes on this way:  Net, to us math people, it is the physics people who are short on the math.<p>Then beyond the math, some of us amateurs have some other issues with physics:<p>Physics in quantum mechanics keeps talking about superposition and linearity.  Us math people know in fine detail just what <i>linearity</i> is -- it&#x27;s central to linear algebra, Banach space, Hilbert space, integration, differentiation, and more.  I have to question if the physics people have a clear understanding of linearity.  Then there is superposition -- tough even to make a mathematically credible guess just what physics means by superposition.<p>The last time I was reading from Feynman&#x27;s <i>Lectures on Physics</i> I recall, before I wanted to get my money back, that he claimed that a particle of unknown position had probability distribution uniform over all of space.  WRONG!  It&#x27;s a trivial argument, maybe even at the middle school level, to show that no such distribution can exist.<p>Then setting aside the math and going for something like actual physics, a particle goes through a beam splitter such as the one in the Mach-Zehnder interferometer.  On one path from the beam splitter there is a detector.  We send in one particle.  The beam splitter yields the wave function in two physically separated parts on two very different paths; we believe this from our experience with the full Mach-Zehnder interferometer.<p>This trial we get a signal from the detector.  Okay, but what happens to the other part of the wave function?  The quantum mechanics lectures and texts don&#x27;t say.<p>Professor Hossenfelder, some of us amateur physics people are actually math people (e.g., with a Ph.D. degree from a world famous research university and have published peer-reviewed original research in math with theorems and proofs) and look at physics people as very short on math and also sometimes even their physics.  Uh, for $50 an hour, I will tutor you in linearity, Lebesgue theory, Hilbert space, and probability theory (based on sigma algebras).