Applications of

Quantum Mechanics

Quantum Mechanics

Implications of

Quantum Mechanics

Quantum Mechanics

39. Quarks. Elementary Particle Systematics. Strings.

Summary

This is a compendium of concepts that are important in current physics but are not particularly relevant to the question of interpretation. Properties of elementary particles, string theory and supersymmetry are reviewed.

Over the last few decades, the list of ‘elementary’ particles has grown from a few—electrons, neutrinos, protons, neutrons, photons—to several dozen. The question then becomes whether all these particles are really just combinations of some small set of ‘most elementary’ particles. A good deal of progress has been made on answering this question in the affirmative and we will briefly describe the resulting scheme here. (Note: ‘particle’ here is shorthand for ‘particle-like wave function.’)

Fermions and bosons. First, particles divide into two general groups, depending on their spin. Fermions have spin 1/2, 3/2, …, and bosons have spin 0, 1, 2, …

Three forces, three charges. Historically, the first charge was the electric charge, associated with the electromagnetic force, the force that holds atoms together. Now there is also a ‘weak’ charge associated with the weak force that causes nuclear decay and creates neutrinos, . And there is a ‘strong’ charge, associated with the strong quark-quark forces that hold protons, neutrons and nuclei together.

Quarks. Color. Fermions divide into two general groups, with the lighter ones (smaller mass) called leptons and the heavier ones baryons. All leptons have strong charge zero. All baryons contain quarks, which have strong charge 1. No one has ever seen an isolated quark; all the particles we can detect are composed of either three quarks or of a quark and an anti-quark bound together. Each quark comes in three ‘colors,’ (an arbitrary designation) say red, yellow and blue. Their existence is inferred from the systematics and force properties of the baryons.

Generations of fermions. The fermions fall into three generations of particles. All three generations have exactly the same charges and force properties, but they have successively higher masses.

First generation:The up quark has electromagnetic charge of +2/3 while the down quark has charge -1/3. A proton is composed of two up quarks and one down quark while a neutron is composed of two downs and one up.

Up and down quarks, electron, and electron neutrino.

Second generation:

Strange and charmed quarks, muon (a fat electron), and muon neutrino.

Third generation:

Bottom and top quarks, tau (a very fat electron), and tau neutrino.

Each quark comes in three colors.

Only the first generation occurs in ‘everyday’ situations.

The neutrinos were originally thought to have zero mass, but are now presumed to have very small masses.

Antiparticles. Each of the fermions has an antiparticle. It has the same mass as the original particle but opposite charges. A particle and its antiparticle can annihilate, leaving only vector bosons. For example, an electron and a positron can annihilate each other, leaving only two photons.

Interaction-mediating vector bosons. The fermions (electrons and so on) interact in a peculiar way. One fermion will create a spin 1 (vector) boson (

*why*this happens is not understood), the boson will travel through space, and then annihilate at another fermion, transferring energy and momentum from the first to the second fermion, and thereby

*mediating*the interaction. The interaction-mediating boson for the electromagnetic interactions is the familiar photon. For the weak interaction, there are three, the W

^{+}, the W

^{-}, and the Z

^{0}. And for the strong interaction there are eight interaction-mediating vector bosons called gluons.

Mass. Higgs. Initially, all particles are thought to have zero mass. But a ‘flaw,’ an asymmetry, develops in the vacuum state and this flaw gives mass to all particles except the photon and gluons. The flaw is usually thought to take the form of a particle, called the Higgs particle. It is hoped that the existence of the Higgs will soon be experimentally confirmed.

The vacuum state. What exists in ‘empty’ space, where the usual particle-like wave functions are zero? In modern physics, it is not just ‘nothing.’ Instead, it is a seething soup of particle-like wave functions going in and out of existence. The vacuum is thought of as a very dense state of fermions and interaction-carrying vector bosons with a ‘density’ of something like (L

_{p})

^{-3}where L

_{p}˜10

^{-35}m is Planck’s length. The fermions are continuously creating and annihilating bosons and so, on the scale of the Planck length, the vacuum is not ‘smooth;’ there are vacuum fluctuations. The Casimir effect, in which two very smooth plates are put close together, shows that the vacuum state can produce observable effects.

Quantum field theory (QFT). Because particles can be created and destroyed (annihilated), the original, simple Schrödinger form of quantum mechanics, with a constant number of particles, becomes awkward. Thus it is useful to recast the formulation of quantum mechanics in terms of

*operators*(the ‘field’ operators of quantum field theory) that create and annihilate particles.

Fermion operators ‘anti-commute’ while boson operators commute. The anti-commutation of the fermion operators means that two electrons can never be in the same state in an atom, and two protons (or two neutrons) can never be in the same state in a nucleus. In fact, the stability of matter depends critically on this anti-commutation property.

Strings, supersymmetry, more than three space dimensions. Aside from interpretive questions, there are two major problems with quantum mechanics. One is that many calculations give infinity (this problem can be circumvented but not eliminated) and the other is that it has proved extremely difficult to unify quantum mechanics and gravity. These three concepts—strings, supersymmetry, and 7, 8 or more dimensions to space—are attempts to solve these two problems. None of these three concepts is particularly relevant to the interpretive problem.

Gravity. (See Ref. 4)For various reasons, it has proved very difficult to integrate gravity into quantum mechanics. I would like to suggest one possible way of doing this (even though I am an amateur in this aspect of physics) which integrates gravity into quantum mechanics in a somewhat natural way.

First, one can make a general argument that gravity should be directly connected with quantum mechanics. Gravity is, in a sense, a theory about mass. But mass is a child of (special) relativistic quantum mechanics because mass follows, via group representation theory, from the Lorentz invariance and linearity of quantum mechanics. In addition the Higgs boson, which also has its natural habitat in (special) relativistic quantum mechanics, is presumed to give mass to particles. So we would be a little surprised if there were not a close connection between gravity and “conventional” quantum mechanics.

How might this be implemented? We know that particles—quarks, electrons and so on—polarize the vacuum; that is, they affect the microscopic structure of the vacuum. So it is conceivable that a macroscopic concentration of particles might slightly change the macroscopic structure of the vacuum. For example, it might slightly change the density of particlelike functions that make up the vacuum. The smallness in the change of the structure of the vacuum could account for the smallness of the gravitational constant.

One need not solve the whole vacuum problem to set this up. One probably only needs certain properties of the “single-particle” density matrix, perhaps the local energy-momentum density. That is, the proposal is that gravity is a

*macroscopic*theory of the vacuum in the presence of concentrations of matter.

Space-time. How would one derive the gravitational equations? I am not certain. But it would presumably have something to do with the way in which a space-time grid is superimposed on the vacuum. In a vacuum, when there are no concentrations of matter, the vacuum state is invariant under the P

_{µ}and the µx

_{µ}are defined by . But when there are concentrations of matter then (presumably) the vacuum state is not invariant under the “local” space-time generators and, very roughly, the x

_{µ}are defined by something like . From this assumption about the properties of the vacuum, I conjecture that one could derive the general relativistic equations connecting space, time, and densities of matter.

Note that the local P

_{µ}has two roles; it generates the local x

_{µ}, and its expectation value is proportional to the local energy-momentum density, including the change in local energy-density of the vacuum.

This scheme, where space and time are “emergent”—derived from how the momentum operators act on the vacuum—rather than “fundamental” properties may make it easier to conceptually understand how matter could alter space and time.