IIIA2. The Particlelike Properties of Matter.

1. Introduction. 

The basic description of particles, used since 1700 and even before, is that they are localized carriers of the particlelike properties of mass, energy, momentum, angular momentum or spin, and charge.  What we will mathematically show here, however, is that the wave function possesses these particlelike properties.  That is, even if there are no particles, the particlelike attributes, with their usual addition and conservation laws, will still be properties of the wave function.  Thus these particlelike attributes provide no evidence for particles because the physical universe described by the wave function alone already possesses them; the concept of particles would be redundant in this respect.

2. Group Representation Theory. 

Group representation theory is most important in quantum mechanics (see sections IIB.7 and IVA).  It implies that if a linear equation is invariant under certain operations, then the solutions to the equation will have certain properties.

The Hydrogen Atom

This idea is most easily illustrated using the hydrogen atom, with Schrödinger equation

 (IIA2-1)  

If we change to a new coordinate system, 

(IIA2-2)  

then both r and are invariant; 

(IIA2-3)  

Thus the form of the equation in the new coordinate system is exactly the same as the form in the old system; that is, the equation is invariant under rotations.  This implies that if is a solution, then so is .

Classifying Solutions.

Group representation theory gives us a way of classifying solutions, using the generators of infinitesimal transformations.  For rotations, these are

 (IIA2-4)  

 These (aside from constants) are the angular momentum operators.  The total angular momentum is

(IIA2-5)  

We can then choose solutions of the Schroedinger equation so they are eigenfunctions of  and .  The eigenvalues classify the solutions.  All solutions to the Schrödinger equation are then linear combinations of these eigenfunction solutions, classified according to their angular momentum.

Physical Properties of Solutions.

The values of  and  are, however, not just a means of classifying solutions.  They also correspond to physical properties of the solutions.  For example, if a spin 1 particle is put through a Stern-Gerlach apparatus, then each of the three different solutions, corresponding to , will take a different trajectory.  Thus in general the group representation labels correspond to physical characteristics of the wave functions they label.

3. Relativistic Quantum Mechanics.
Mass, Energy, Momentum and Spin. 

Group representation theory can be applied to relativistic quantum mechanics, in which the linear equations are invariant under the four dimensional space-time “rotations@ of the homogeneous Lorentz group, and under space-time translations.  In 1939, E. P. Wigner proved that the solutions—that is, the wave functions—of the relativistically invariant equations of quantum mechanics can be classified according to their mass, energy, momentum and spin. That is, the wave functions can be classified according to four of the major classical properties of particles.

Further, not only can wave functions be classified in this way, but it also follows from representation theory that these particlelike properties of the wave function obey the proper addition rules; energy, momenta and the z-component of spin add algebraically while the total angular momentum adds in a more complicated way.  In addition, the mathematics also implies that total energy, momentum and angular momentum will be conserved.  And finally, the equations of motion imply that these labels correspond exactly, quantitatively, to the observed physical properties of the system corresponding to the wave function.

Thus, the Wigner theorem implies that it is logically consistent to assume it is the wave function that carries or possesses the particlelike properties of mass, energy, momentum and spin.  That is, no prediction regarding these quantities is lost or contradicted by experiment if it is assumed they are properties of the wave function rather than particles.

One might be tempted to dispute our interpretation of the Wigner results by saying that the particlelike properties of the theorem refer to the properties of the “particle associated with” the wave function rather than to the properties of the wave function itself.  However, the concept of particles never enters the Wigner derivation; the results follow strictly from the group-theoretic structure of the relativistic equations for the wave function.  So this associated-particle interpretation of the Wigner results is not supported in the slightest by the mathematics (or by any reasoning or observation that I know of).

4. Charge. 

Over the past few decades, the idea of an internal symmetry group—one not related to rotations or translations in space and time—has developed.  (The most commonly used groups are U(1)XSU(2) for electroweak interactions, SU(3) for strong interactions, and SU(5) for unified theories.)  Again, because the quantum mechanical equations are linear, group representation theory can be applied.  In this case, it is the charges that classify the solutions (with the solutions being the wave functions).  And it follows from representation theory that the three charges, strong, electromagnetic and weak, obey the addition and conservation laws we expect.  Thus the particlelike property of charge can also be considered to be a property of the wave function.   

5. Summary.

All the primary properties we have associated with particles for three hundred years—mass, energy, momentum, spin and charge—can be attributed to the wave function without loss of predictive power.  The rigorous derivation of these classical particlelike properties from quantum mechanics is a most remarkable result!  And since these properties have been a major reason for keeping the concept of particles, this result immediately undermines our confidence in that concept.

6. A Note on Nomenclature. 

Even though there are in all probability no particles, it is often convenient to use the term “particle” when describing a certain physical system.  So when we use the word “electron,” what we mean is a wave function with the measured mass, , of  “the electron,” spin ½, electromagnetic charge –e, weak charge –½, and strong charge 0.  Similarly, a photon refers to a photonlike wave function that has mass 0, spin 1, and strong, weak and electromagnetic charges of 0.  But when we use the words “electron” or “photon” we should be careful not to conjure up in our minds a point particle.

7. The Correctness of Quantum Mechanics. 

This very tight interlock between linearity, relativity, the internal symmetry group representation theory, and the classical, well-established particlelike properties is probably as close as one can get to a proof that the basics of quantum mechanics will never be superseded or found to be incorrect.  (See, however, section IV, for a proposed theory that underlies current quantum mechanics but does not supersede the basic principles of linearity, relativistic invariance and internal symmetry groups).