Transcending the impasse, part IV

Planck’s constant

It all started with the work of Max Planck. He famously introduced the notion that the energy absorbed or emitted during an interaction is proportional to the frequency of the field being absorbed or emitted. The proportionality constant h is today considered as a fundamental constant of nature. In honor of Max Planck is called Planck’s constant.

Max Planck, the father of quantum mechanics

The reason why we need to look at the Planck constant for transcending the impasse in physics is because there seem to be some confusion as to the role that it plays in quantum mechanics. The confusion manifests in two aspects of quantum mechanics.

One of these aspects is related to the transition from quantum to classical physics, which we have considered before. It is assumed that one should recover classical physics from quantum physics by simply taking the limit where Planck constant goes to zero. Although this assumption is reasonable, it depends on where the constant shows up. One may think that the presence of Planck’s constant in expressions should be unambiguous. That turns out not to be the case.

An example is the commutation relation for spin operators. Often one finds that the commutator produces the spin operators multiplied by Planck’s constant. According to this practice the limit where Planck’s constant goes to zero would imply that spin operators must commute in the classical theory, which is obviously not correct. Spin operators are the generators of three-dimensional rotations which still obey the same algebraic structure in classical theories as they do in quantum theories.

So when should there be a factor of Planck’s constant and when not? Perhaps a simple way to see it is that, if one finds that a redefinition of the quantities in an expression can be used to remove Planck’s constant from that expression, then it should not be there in the first place.

Using this approach, one can consider what happens in a Hamiltonian or Lagrangian for a theory. Remember that both of these are divided by Planck’s constant in the unitary evolution operator or path integral, respectively. One also finds that the quantization of the fields in these theories always contains a factor of the square root of Planck constant. If we pull it out of the definition and make it explicit in the expression of the theory, one finds that Planck’s constant cancels for all the free-field terms (kinetic term and mass term) in the theory. The only terms in either the Hamiltonian or the Lagrangian where the Planck constant remains are the interaction terms. This brings us full circle to the reason why Max Planck introduced the constant in the first place. Planck’s constant is specifically associated with interactions.

So if one sets Planck constant to zero in a theory, the result is that it removes all the interactions. It leads to a free-field theory without interactions, which is indistinguishable form a classical theory. Interactions are responsible for the changes in the number of particles and that is where all the quantum effects come from that we observe.

The other confusion about Planck’s constant is related to the uncertain principle. Again, the role that Planck’s constant plays is that it relates two quantities that, on the one hand, is the conjugate variable on phase space with, on the other hand, the Fourier variable. Without this relationship, one recovers the same uncertainty relationships between Fourier variables in classical theories, but not between conjugate variables in phase space. Planck’s relationship transfers the uncertainty relationship between Fourier variables to conjugate variables on phase space. So, the uncertainty relationship is not a fundamental quantum mechanical principle. No, it is the Planck relationship that deserves that honor.

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