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Equal footing in quantum physics

So, regarding interaction of matter, there’s a major inversion of perspective between classical physics and quantum field theory (QFT): hallmarked particles which create fields vs. excitations created (and destroyed) in ubiquitous extant fields. As Ethan Siegel said:

… in quantum field theory, quantum fields aren’t generated by matter. Instead, what we interpret as “matter” is itself a quantum field. [1]

In classical physics, matter particles and fields are not on equal footing. Fields are derived from particles. In classical physics, particles set the stage. In QFT, fields are the stage: localized excitations interact and transfer energy and momentum (linear and angular).

In QFT, something similar might be said for field energy and vacuum energy – a case for equal footing as energy counterparts.

And for those energy counterparts, Sean Carroll points out the change in perspective from a landscape of position and velocity (x,v) where momentum (p) is secondary to one where x and p are on more of an equal footing – treated almost symmetrically. [2]

And, as said many times, physics is all about symmetry.

Wiki: In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article More is Different that “it is only slightly overstating the case to say that physics is the study of symmetry.”

Notes

[1] Science communicator and Forbes contributor Ethan Siegel discusses the QFT tally in this Forbes article, “Ask Ethan: Are Quantum Fields Real?” (Nov 17, 2018), which contains some useful visuals.

In physics, a field, in general, describes what some property of the Universe is everywhere in space. It has to have a magnitude: an amount that the field is present. It may or may not have a direction associated with it; some fields do, like electric fields, some don’t, like voltage fields. When all we had were classical fields, we stated that the fields must have some kind of source, like particles, which results in the fields existing all throughout space.

Have a particle and antiparticle annihilating? That’s described by equal-and-opposite excitations of a quantum field. Want to describe the spontaneous creation of particle-antiparticle pairs of particles? That’s also due to excitations of a quantum field.

Every particle in the Universe, as we understand it, is a ripple, or excitation, or bundle-of-energy, of the underlying quantum field.

it’s more accurate to view the entire Universe as a complicated quantum field that, itself, contains all of physics. Quantum fields can describe an arbitrarily large number of particles that interact in all ways our theories can conceivably allow. And they do this not in some vacuum of empty space, but amidst a background of not-so-empty-space, which plays by the rules of QFT, too.

[2] See Sean Carroll’s video chat “The Biggest Ideas in the Universe | 4. Space” (Apr 14, 2020) on his YouTube channel.

The Hamiltonian way is different. the Hamiltonian way says look at this structure that we have, a phase space here. okay X and P. put X and P on more of an equal footing. so rather than thinking of the momentum as something that is derived from position by taking the velocity then multiplying by the mass, P = MV. this is the usual way of thinking about it. rather than doing that, say that the position and the momentum are two truly independent things. okay, and construct from these independent things a function of them, a function of phase space, called the Hamiltonian. [H(x,p)]

In the Hamiltonian, you know, it’s basically the energy … it’s the energy as a function of position and momentum. that’s the whole point – that it’s not a function of position and velocity, it’s a function of position and momentum.

There’s no mention of velocity here. these two equations are on an equal footing, and in fact you can derive these two equations from this Hamiltonian function.

So why am I telling you this, right. this is that question will appear many many times in these videos. why am I telling you this, why are we bothering with all this formalism? well, the answer is because these equations and this picture really do put P and X on kind of an equal footing, right. they treat them almost symmetrically.

One thought on “Equal footing in quantum physics

  1. Field-ly

    there were forces and objects in Newton’s day,
    with x & V the interplay.
    but now we have the Hamiltonian way,
    and x & P get equal play.

    [chorus]
    the Hamiltonian way, the Hamiltonian way,
    x & P get equal play

    in space we moved from x at V,
    as forces produced all the to & fro.
    P derived secondarily,
    matter has inertia, as you know.

    [chorus]

    so long the particular way used to be
    mass times velocity to get the P.
    more independent things are visionary,
    as a phase space function of x & P.

    [chorus]

    look at space in a new perspective,
    velocity placed as restrospective.
    an equal footing, basically,
    in physics we study symmetry.

    now it’s all about energy, of a path,
    with equations of field theory, do the math.
    the Hamiltonian way, the Hamiltonian way,
    hey, hey!

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