Why do different wave equations have varying numbers of derivatives?

[DiracPool]

In the early days of quantum mechanics, Erwin Shroedinger was developing his famous wave equation. I may need to double check this, but I believe he initially tried to develop a relativistic wave equation but essentially came up with a prototype of the Klein-Gordon equation and abandoned it. Klein-Gordon actually developed the equation and apparently it doesn't work for the electron but it works for spin-0 bosons, such as the Higgs.

Shroedinger went on to develop his famous wave equation, which ostensibly contains all the information you would ever want to know about a particle/system, and if you're really interested, you can just apply a measurable operator to that equation and find your position, momentum, or energy.

My question here regards the structure of these equations. Klein-Gordon has two derivatives of time and two derivatives of space. This is the logical first stab at formulating a relativistic wave equation just looking at the Einstein relation, E^2=(pc)^2+(mc^2)^2.

The cononical Shroedinger equation has one derivative of time and two derivatives of space.

And the Dirac equation has one derivative of time and one derivative of space.

So again, my question is basically this...How can we be so cavalier about differentially differentiating these wave equations? Is there any larger model of quantum mechanics that can justify arbitrarily taking different time and space derivatives for these three equations?

 

[strangerep]

[...] my question is basically this...How can we be so cavalier about differentially differentiating these wave equations? Is there any larger model of quantum mechanics that can justify arbitrarily taking different time and space derivatives for these three equations?

It's not at all cavalier.

The KG and Dirac equations are for the relativistic case, where the basic symmetry group is the Poincare group.

The Schroedinger eqn is for the nonrelativistic case, where the basic symmetry group is the Galilei group.

In both cases, the underlying principle is that we want wave equations that characterize a particular representation of the relevant symmetry group. E.g., the KG equation is for Poincare reps that are massive, with spin 0. The Dirac equation is for Poincare reps that are massive, with spin 1/2. The (ordinary) Schroedinger equation is for Galilei reps that are massive, with spin-0. (There's also a "Pauli eqn" for spin 1/2 in the latter case.)

I don't know whether you've encountered the concept of "group representation" already, upon which the above depends. If not, then I guess a much longer answer is needed.

 

[DiracPool]

In both cases, the underlying principle is that we want wave equations that characterize a particular representation of the relevant symmetry group.

So we have to taylor make "designer" wave equations for different symmetry groups? Seems a little unparsimonious, doesn't it? What is the logic in the relativistic Dirac equation that has one space derivative that recovers the non-relativistic Shroedinger equation with two space derivatives? I don't get it. And then you have the KG equation that works for "specialty" conditions such as zero spin particle fields that we feed in two time derivatives?

 

[strangerep]

So we have to taylor make "designer" wave equations for different symmetry groups? Seems a little unparsimonious, doesn't it?

"Taylor make"? I guess you mean "tailor make"?

But no, the details of the symmetries determine the wave equations methodically. That's what (infinite-dimensional) representation theory is all about. It's quite a large, and fairly well-developed, subject.

The only "tailoring" is that one must put in causality by hand -- see below.

What is the logic in the relativistic Dirac equation that has one space derivative that recovers the non-relativistic Shroedinger equation with two space derivatives? I don't get it.

It's essentially the same technique that recovers Newtonian mechanics from Einsteinian mechanics in the low velocity limit. The high-fallutin term is "group contraction" -- by letting $v/c \to 0$, one can "contract" the Poincare group to the Galilei group. Applied directly to the Dirac equation, one gets the Pauli equation in that limit.

And then you have the KG equation that works for "specialty" conditions such as zero spin particle fields that we feed in two time derivatives?

The KG equation is just based on the formula for the mass$^2$. It is applicable to everything, except that it has this negative energy problem, and hides some of the detail for nonzero spin. The correct symmetry group to use is really the full Poincare group (including parity transformations), restricted by causality. (Strictly speaking, this is a semigroup, since we only want forward time evolution, with energy bounded below.) This is how modern quantum field theories are constructed: by finding causal representations of the full Poincare group. (Ref: Weinberg vol 1.)

 

[DiracPool]

"Taylor make"? I guess you mean "tailor make"?

Lol. Ok, strangerep, you busted me. I must have been diverted by my golf visor hanging up in my closet..

I guess I have to do some research into these symmetry groups, thanks for the lead.

 

[bhobba]

I guess I have to do some research into these symmetry groups, thanks for the lead.

Read chapter 3 - Ballentine - Quantum Mechanics - A Modern Development.

Thanks
Bill