QM (Quantum Mechanics) from SR (Special Relativity)

A physical derivation of Quantum Mechanics (QM) using only the assumptions of Special Relativity (SR) as a starting point...

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The following is a derivation of QM-Quantum Mechanics from SR-Special Relativity. It basically highlights the few assumptions necessary to generate QM given SR as the base assumption.

A lot of the texts on quantum mechanics that I have seen start with a few axioms, most of which are non-intuitive, and don't really seem to be related to anything in classical physics.  These assumptions then build up to the Schroedinger equation, at which point we have quantum mechanics.  In the more advanced chapters, the books will then say that we need a wave function that obeys special relativity.  They then proceed by positing the Klein-Gordon equation, saying that it is the relativistic version of the Schroedinger equation.  It is then shown that quantum mechanics agrees very nicely with all of the requirements of special relativity, and that in fact, things like the spin-statistics theory come from the union of quantum mechanics and special relativity.

But, one facet of quantum theory that has always intruiged me is this: quantum mechanics seems to join up very well with special relativity, but not with general relativity.  Why?

Thinking along that line led me to the following ideas:  Why do the textbooks start with Schroedinger, and then say that Klein-Gordon is the relativistic version? What if quantum mechanics can actually be derived from special relativity? If so, then one can more correctly state that Schroedinger is actually the low-velocity/low-energy limit of Klein-Gordon, just as Newtonian physics is the low-velocity limit of Relativistic physics. Can you get quantum mechanics without using the standard QM axioms?  Can the axioms themselves actually be derived from something that makes a little more sense?  Newton's laws can be derived as low-energy limits of special relativity. So, starting with SR, and it's two simple axioms, what else do you actually need to get QM?

Also, if it turned out that QM can be derived from SR, that would sort of explain that difficulties of making it join up with GR.  If quantum theory is derivable from a "flat" Minkowski space-time, then GR curvature effects are something above and beyond QM.

Since the language of SR is beautifully expressed using 4-vectors, I will use that formalism. There are quite a few different variations of 4-vectors that can correctly describe SR. I use the one that has only real (non-complex) notation throughout SR. The imaginary unit ( i ) is introduced only at the last step, which gives QM. As you can note from the outline, there are only a few steps necessary. By the way, SR is an excellent approximation for the majority of the currently known universe, including on the surface of Earth.  It is only in the regions of extreme curvature, such as near a stellar surface or black hole, that GR is required.

See the 4-Vectors Reference for more reasoning on the choice of notation, and for more on four-vectors in general.
 


Interesting points include:

*All particles, massive or massless, have a 4-velocity magnitude of c, the speed of light.
*A number of particle properties are simply constants times another property
*Wave-particle duality appears to occur purely within SR - ex. Relativistic Optics, Relativistic Doppler Effect.
*QM is apparently generated simply by allowing the particles to move in imaginary/complex spaces within spacetime.
*The Quantum Superposition Principle, usually assumed as axiomatic, is apparently a consequence of the Klein-Gordon equation being a linear wave PDE.

Notation and Properties of SR 4-Vectors

guv = guv = DiagnolMatrix[1,-1,-1,-1] Minkowski Spacetime Metric: This is the "flat" spacetime of SR
A = Au = (at,ax,ay,az) = (a0,a1,a2,a3) = (a0,a) A typical 4-vector
Au = (at,-ax,-ay,-az) = (a0,-a1,-a2,-a3) = (a0,-a) A typical 4-covector; we can always get the 4-vector form with Au = guvAv
A·B = guv Au Bv = Av Bv = Au Bu = +a0b0-a1b1-a2b2-a3b3 = +a0b0-a·b The Scalar Product relation

Useful Quantities

γ[v] = 1 / Sqrt[1-(v/c)2] : Lorentz Scaling Factor (gamma factor)
τ[v,t] = t / γ : Proper Time
Sqrt[1+x] ~ (1+x/2) for x<<1 : Math relation often used to simplify Relativistic eqns. to Newtonian eqns.

Fundamental/Universal Physical Constants (Lorentz Scalars)

c = Speed of Light
hbar = h/2π = ℏ = Planck's Reduced Const aka. Dirac's
mo = Particle Rest Mass(varies with particle type)

Fundamental/Universal Physical 4-Vectors (Lorentz Vectors)

(this notation always places the c-factor in the time-like part, and the name goes with the space-like part)

4-Position R = (ct,r)
4-Velocity U = γ(c,u)
4-Momentum P = (E/c,p) = moU = γ mo(c,u)
4-CurrentDensity J = (cρ,j) = γ ρo(c,u)
4-WaveVector K = (ω/c,k) = 1/hbar (E/c,p)
4-Gradient ∂ = (∂/c∂t,-del) = (∂t /c,-del)
4-EM VectorPotential AEM = (ΦEM/c, aEM)

Fundamental/Universal Relations

U = dR/dτ "4-Velocity is the derivative of 4-Position wrt. proper time"
P = moU
J = ρoU
K = 1/hbar P
∂ = -iK "This relation is usually written as a correspondence ∂ ~ -iK , rather than a pure equality"
D = ∂ + iq/hbar AEM "where AEM is the EM vector potential and q is the EM charge"

Derived Physical Constants (Scalar Products of Lorentz Vectors give Lorentz Scalars)

R·R = (Δs)2 = (ct)2-r·r = (ct)2-|r|2
U·U = (c)2
P·P = (moc)2 : J·J = (ρoc)2
K·K = (moc/hbar)2
∂·∂ = -(moc/hbar)2   "This relation may require a bit more justification, but it works..."
D·D = -(moc/hbar)2   "The same KG equation, but with minimal coupling to an EM potential"


Now then, how do we get QM out of SR?

Start with a special relativistic spacetime for which the invariant interval is given by R·R = (Δs)2 = (ct)2-r·r = (ct)2-|r|2.
This is just a "flat" Euclidean 3-space with an extra, reversed-sign dimension, time, added to it.
This interval is Lorentz Invariant. Space-like intervals are (-)negative, time-like intervals are (+)positive, and light-like intervals are (0)null.
One can say that the universe is the set of all possible events in spacetime.

Events/Particles:

Now, let's examine the interesting events.
Let there exist particles (which can carry information) that move about in this spacetime.
Each particle has a location at an event (a time and a place) 4-Position R = (ct,r).
The factor of c is inserted in the time part to give the correct, consistent dimension of length to this 4-vector.
In fact, every SR 4-vector has this constant c-factor to give consistent dimensions.
A particle is simply a self-sustaining event, or more correctly a worldline of connected events, which "carries" information.
The information that a particle can "carry" include mass, charge, any of the various hypercharges, spin, polarization, phase, frequency, energy, etc..
These are the particles' properties.

Motion/Dynamics:

Let these particles can move around within the spacetime.
The 4-Velocity of an event is given by dR/dτ, or the total derivative of the 4-Position with respect to its Proper Time.
This gives the 4-Velocity U = γ(c,u) where γ(v) = 1 / Sqrt[1-(v/c)2].
This particle, if its rest mass m0>0, moves only in the direction of +time along its own worldline Uworldline = (c,0).
Interestingly, all stationary (v = 0) massive particles move into the future at c, the Speed of Light;
If the particle has rest mass mo = 0, it moves in a null or light-like direction. This is neither along time nor along space, but "between" them.
These light-like particles, with a v = c, have a 4-Velocity: Ulight-like = Infinite c(1,n), where n is a unit space vector.
Since this is rather undefined, we will use the 4-Wave Vector, introduced later, to describe photons.
A particle only has a spatial velocity u with respect to another particle or an observer.
We have the relation Sqrt(U·U) = c. This says that the magnitude of the 4-velocity is c, the speed of light. This result is general, massive or massless!
What all this means is that all light-like particles live on the "surface" null-space of the Light Cone, between time and space,
while all massive particles live within the "interior" the Light Cone.

Light Cone


        | time-like interval(+)

                      / light-like interval(0)
worldline

       |
       |        c
\   future /
  \    |    /
    \  |  /         -- space-like interval(-)
      \|/now
      /|\
    /  |  \         elsewhere
  /    |    \
/   past    \
       |        -c

Mass/Energy/Momentum:

One of the basic properties of particles is that of mass. Each particle has a rest mass mo.
This mass, along with the velocity of a particle, gives 4-Momentum P = moU.
Nature seems to indicate that one of the fundamental conservation laws is the Conservation of 4-Momentum.
This comes from the idea that a system remains invariant under time or space translations in an isotropic, homogeneous universe.
The sum of all particle 4-Momenta involved in a given interaction is constant; it has the same value before and after a given interaction.
The 4-Momentum relation P = moU gives 4-Momentum P = (E/c,p) = moU = γmo(c,u).
This gives the Einstein Mass-Energy relation, E = γ moc2, or E = mc2 where m = (γ mo).
Note that for light-like particles, the result using this formula is undefined since Elight-like = Infinite 0 c2.
Presumably, the m = (γ mo) factor must scale in some way (i.e. like a delta function) to give reasonable results.

Waves:

The energy of light-like particles can be obtained another way.
It turns out that every photon (light particle) has associated with it a 4-WaveVector K = (ω/c,k), where ω = temporal angular frequency.
Through the efforts of Planck, Einstein, and de Broglie, it was discovered that K = ( 1 / hbar )P = (ω/c,k) = 1/hbar (E/c,p).
Planck discovered hbar based on statistical-mechanics/thermodynamic considerations.
Einstein applied Planck's idea to photons to give E = hbar ω.
de Broglie realized that every particle, massive or massless, has p = hbar k.
Putting it all together gives P = hbar K = (E/c,p) = hbar(ω/c,k).
Note also that the 4-WaveVector (a wave-like object) is just a constant, hbar, times the 4-Momentum (a particle-like object).
This means that photons, or other massless quanta, can act like localized particles and massive quanta can act like non-localized waves.
That gives the Mass-Energy relation for all kinds of particles, ( E = γ moc2 = hbar ω ), and also gives the relation for m = (γ mo) = ω hbar/c2.
This leads into the wave-particle duality aspect of nature, and we haven't even gotten to QM yet!

Note: "There is a duality of particle and wave even in classical mechanics, but the particle is the senior partner, and the wave aspect has no opportunity to display its unique characteristics." - Goldstein, Classical Mechanics 2nd Ed., pg 489 (The relation between geometrical optics and wave mechanics using the Hamilton-Jacobi Theory).

I need to emphasize here that the 4-WaveVector can exist as an entirely SR object (non-QM). It can be derived in terms of periodic motion, where families of surfaces move through space as time increases, or alternately, as families of hypersurfaces in spacetime, formed by all events passed by the wave surface. The 4-WaveVec is everywhere in the direction of propagation of the wave surfaces. From this structure, one obtains relativistic/wave optics, without ever mentioning QM.

When Planck was working on the blackbody problem, he discovered that classical statistical mechanics requires the existence of h (but does not define its value). When Einstein studied the photoelectric effect, he then found E = hbar ω, which related energy E with temporal angular frequency ω. This is a particle effect, from which Einstein concluded that photons were actually particle quanta. Since this equation is the time component of a 4-vector, the space component logically follows, giving the de Broglie relation p = hbar k. Thus, wave-particle duality enters the picture at this stage, but we still haven't mentioned QM yet.


 I believe that there is more to the 4-WaveVec than other people have figured on (i.e. more importance to the overall phase of the waves). More on that later...
Also, the question always arises: What is waving? I assume that it is simply an internal property of a particle that happens to be cyclic. This would allow all particles to be "waves", or more precisely to have a period, without the need for a medium to be waving in. Also, note that the phase of the 4-WaveVec was not defined. Presumably, (2π) of 4-WaveVec's could have the same K 4-Vector.  However, another interpretation could be the symmetry between 4-vectors and One-Forms, where the 4-vectors consist of "arrows" and one-forms consist of "parallel lines".  The length of arrow along the lines is the dot-product operation, which results in a Lorentz scalar number.
Also, it is at this step that I believe a probabilistic description is being imposed on the physics.

Spacetime Structure:

Now, let's get to the really tough stuff.
There is a thing called the 4-Gradient ∂ = (∂/c∂t,-del) = (∂t /c,-del), where ∂ is the partial derivative function.
It tells you about the changes/variations in the "surface" of spacetime.
This 4-vector is significantly different from the others. It is a function that acts on a value, not a value itself.
It also has a negative sign in the space component, unlike the other "physical type" vectors.
I must admit that I don't have a good intuitive feeling for this object, but it gives the correct physics.

When it is applied to the 4-CurrentDensity, it leads to the Conservation of Charge equation.
∂·J = ∂/c∂t(cp)+del·j = ∂p/∂t +del·j = 0.
This says that the change in charge-density with respect to time is balanced by the divergence or spatial flow of current-density.

The same thing can be applied to particle 4-Momentum:
∂·P = ∂/c∂t(E/c)+del·p = (1/c2)∂E/∂t +del·p = 0. ∂E/∂t+c2del·p = 0.
This says that the change in energy with respect to time is balanced by the divergence or spatial flow of momentum.
In fact, this is the 4-Vector Conservation of Momentum Law.
Energy is neither created nor destroyed, only transported from place to place in the form of momentum.
This is the strong, local form, of conservation - the continuity equation.

Now comes Quantum Mechanics (QM)!

Based on empirical evidence, QM has given the correct calculation/approximation of more phenomena than any other theory, ever, with the exception of enhancements like QED, which are modifications of QM!
We have the following simple relation: ∂ = -i K or K = i ∂.
Again, most books will call this a correspondence, rather than a strict equality, whatever that means.
This innocent-looking, very simple relation gives all of Standard QM, by providing P = hbar K = i hbar ∂.
This gives (E = i hbar ∂/∂t) and (p = -i hbar del).
These are the standard operators used in the Schrödinger/Klein-Gordon eqns (as well as other relativistic quantum field equations), which are the basic QM description of physical phenomena.
This essentially gives the Unitary Evolution Axiom of QM, which governs how the state of a quantum system evolves in time.

Let's summarize a bit:
We used the following relations: (particle/location-->movement/velocity-->mass/momentum-->wave duality-->spacetime structure)
R = (ct,r)  particle/location
U = dR/dτ movement/velocity
P = moU mass/momentum
K = 1/hbar P wave duality
∂ = -iK spacetime structure


By applying the Scalar Product law to these relations, we get:
U·U = (c)2
P·P = (moc)2
K·K = (moc/hbar)2
and, assuming that we can get away with it:
∂·∂ = (-imoc/hbar)2 = -(moc/hbar)2

Let's look at that last equation.
∂·∂ = (∂/c∂t,-del)·(∂/c∂t,-del) = ∂2/c2∂t2-del·del = -(moc/hbar)2,
∂2/c2∂t2 = del·del-(moc/hbar)2

This is the basic Klein-Gordon equation, the relativistic cousin of the Schrödinger equation!
It is the relativistically-correct, quantum wave-equation for spin-0 particles.
We have apparently discovered QM by simply multiplying with the imaginary unit, ( i ).
Essentially, it seems that allowing SR relativistic particles to move in an imaginary/complex space is what gives QM.
At this point, you have the simplest relativistic quantum wave equation.
The principle of quantum superposition follows from this, as this wave equation obeys the linear superposition principle.
The quantum superposition axiom tells what are the allowable (possible) states of a given quantum system.
I believe that the only other necessary postulate to really get all of standard QM is the probability interpretation of the wave function, and that likely is a simply reinterpretation of the continuity equation, ∂·J = ∂/c∂t(cp) + del·j = ∂p/∂t + del·j = 0, where J is taken to be a "particle" current density.


The Klein-Gordon equation is more general than the Schrödinger equation, but simplifies to the Schrödinger equation in the (v/c)<<1 limit.
Also, extensions into EM fields (or other types of relativistic potentials) can be made using D = ∂ + iq/hbar AEM where AEM is the EM vector potential and q is the EM charge,
and allowing D·D = -(moc/hbar)2 to be the more correct EM quantum wave equation.

D·D = -(moc/hbar)2
(∂ + iq/hbar AEM)·(∂ + iq/hbar AEM) + (moc/hbar)2 = 0

let A'EM = iq/hbar AEM
let M = moc/hbar
then (∂ + A'EM)·(∂ + A'EM) + (M)2 = 0

∂·∂ + ∂·A'EM + 2 A'EM·∂ + A'EM·A'EM  + (M)2 = 0
now the trick is that factor of 2, it comes about by keeping track of tensor notation...
a weakness of strick 4-vector notation

let the 4-Vector potential be a conservative field, then ∂·AEM  = 0
(∂·∂) + 2(A'EM·∂) + (A'EM·A'EM)  + (M)2 = 0

expanding to temporal/spatial components...
( ∂t2/c2-del·del ) + 2(φ'/c ∂t/c - a'·del ) + ( φ'2/c2- a'·a' )  + (M)2 = 0

gathering like components
( ∂t2/c2 + 2φ'/c ∂t/c +  φ'2/c2 ) - (del·del  + 2 a'·del  +  a'·a' ) + (M)2 = 0
( ∂t2 + 2φ'∂t +  φ'2 ) - c2(del·del  + 2 a'·del  +  a'·a' ) + c2(M)2 = 0
( ∂t + φ' )2 - c2(del + a' )2 + c2(M)2 = 0

multiply everything by (i hbar)2
(i hbar)2( ∂t + φ' )2 - c2(i hbar)2(del + a' )2 + c2(i hbar)2(M)2 = 0

put into suggestive form
(i hbar)2( ∂t + φ' )2 = - c2(i hbar)2(M)2 + c2(i hbar)2(del + a' )2
(i hbar)2( ∂t + φ' )2 =  i2c2(i hbar)2(M)2 + c2(i hbar)2(del + a' )2
(i hbar)2( ∂t + φ' )2 =  i2c2(i hbar)2(M)2 [1 + c2(i hbar)2(del + a' )2/ i2c2(i hbar)2(M)2 ]
(i hbar)2( ∂t + φ' )2 =  i2c2(i hbar)2(M)2 [1 + (del + a' )2/ i2(M)2 ]

take Sqrt of both sides
(i hbar)( ∂t + φ' ) =  ic(i hbar)(M) Sqrt[1 + (del + a' )2/ i2(M)2 ]

use Newtonian approx  Sqrt[1+x] ~ ±[1+x/2] for x<<1
(i hbar)( ∂t + φ' ) ~  ic(i hbar)(M) ±[1 + (del + a' )2/2 i2(M)2 ]
(i hbar)( ∂t + φ' ) ~  ±[ic(i hbar)(M) + ic(i hbar)(M)(del + a' )2/2 i2(M)2 ]
(i hbar)( ∂t + φ' ) ~  ±[c(i2 hbar)(M) + c( hbar)(del + a' )2/2(M) ]

remember M = moc/hbar
(i hbar)( ∂t + φ' ) ~  ±[c(i2 hbar)(moc/hbar) + c( hbar)(del + a' )2/2(moc/hbar) ]
(i hbar)( ∂t + φ' ) ~  ±[c(i2)(moc) + (hbar)2(del + a' )2/2(mo) ]
(i hbar)( ∂t + φ' ) ~  ±[-(moc2) + (hbar)2(del + a' )2/(2mo) ]

remember A'EM = iq/hbar AEM
(i hbar)( ∂t + iq/hbarφ ) ~  ±[-(moc2) + (hbar)2(del + iq/hbara )2/2mo ]
(i hbar)( ∂t ) + (i hbar)(iq/hbar)(φ) ~  ±[-(moc2) + (hbar)2(del + iq/hbara )2/2mo ]
(i hbar)( ∂t ) + (i2)(qφ ) ~  ±[-(moc2) + (hbar)2(del + iq/hbara )2/2mo ]
(i hbar)( ∂t ) -(qφ ) ~  ±[-(moc2) + (hbar)2(del + iq/hbara )2/2mo ]
(i hbar)( ∂t )  ~  (qφ )±[-(moc2) + (hbar)2(del + iq/hbara )2/2mo ]

take the negative root
(i hbar)( ∂t )  ~  (qφ ) + [(moc2) - (hbar)2(del + iq/hbara )2/2mo ]
(i hbar)( ∂t )  ~  (qφ ) + (moc2) - (hbar)2(del + iq/hbara )2/2mo

call (qφ ) + (moc2) = V[x]
(i hbar)( ∂t )  ~  V[x] - (hbar)2(del + iq/hbara )2/2mo

typically the vector potential is zero in most non-relativistic settings
(i hbar)( ∂t )  ~  V[x] - (hbar)2(del)2/2mo

And there you have it, the Schroedinger Equation with a potential
The assumptions for non-relativistic equation were:
Conservative field AEM, then ∂·AEM  = 0
(del + a' )2/ i2(M)2 = (del + a' )2/ i2(moc/hbar)2 = (hbar)2(del + a' )2/ i2(moc)2 is near zero
i.e. (hbar)2(del + a' )2 << (moc)2, a good approximation for low-energy systems
Arbitrarily chose vector potential a=0
Or keep it around for a near-Pauli equation (we would just have to track spins, not included in this derivation)


Note that the free particle solution ∂·∂ = -(moc/hbar)2 is shown to be a limiting case for AEM = 0.
Again, see the 4-Vectors Reference for more on this.

Now, let's examine something interesting...
∂·∂ = -(moc / hbar)2: Klein-Gordon Relativistic Wave eqn.
∂ = -i/hbar P
∂·(-i/hbar P) = -(moc / hbar)2
∂·(P) = - i (moc)2 / hbar
∂·(P) = 0 - i (moc)2 / hbar
but, ∂·(P) = Re[∂·(P)], by definition, since the 4-Divergence of any 4-Vector (even a Complex-valued one) must be Real
so ∂·(P) = 0 : The conservation of 4-Momentum (i.e. energy&momentum) for our Klein-Gordon relativistic particle.
This is also the equation of continuity which leads to the probability interpretation in the Newtonian limit.

So, the following assumptions within SR-Special Relativity lead to QM-Quantum Mechanics:

R = (ct,r) Location of an event (i.e. a particle) within spacetime
U = dR/dT Velocity of the event is the derivative of position with respect to Proper Time
P = moU Momentum is just the Rest Mass of the particle times its velocity
K = 1/hbar P A particle's wave vector is just the momentum divided by Planck's constant, but uncertain by a phase factor
∂ = -iK The change in spacetime corresponds to (-i) times the wave vector, whatever that means...

Each relation may seem simple, but there is a lot of complexity generated by each level.

It can be shown that the Klein-Gordon equation describes a non-local wave function, which "violates relativistic causality when used to describe particles localized to within more than a Compton wavelength,..."-Baym. The non-locality problem in QM is also the root of the EPR paradox. I suspect that all of these locality problems are generated by the last equation, where the factor of ( i ) is loaded into the works, although it could be at the wave-particle duality equation. Or perhaps we are just not interpreting the equations correctly since we derived everything from SR, which should obey its own relativistic causality.


Let's examine the last relation on a quantum wave ket vector |V>:
∂ = -iK
∂ |V> = -iK |V> which gives time eqn.[∂/c∂t |V> = -iω/c |V>] and space eqn.[-del |V> = -ik |V>]

A solution to this equation is:

|V> = vn e^(-i Kn·R) |Vn>  where vn is a real number, |Vn> is an eigenstate (stationary state)

Generally, |V> can be a superposition of eigenstates |Vn>

            N
|V> = Sum [vn e^(-i Kn·R) |Vn>]
           n = 1

Going back to the 4-wave vector K, I believe that this is the part of the derivation of QM from SR that the quantum probabilistic interpretation becomes necessary. Since the 4-wave vector as given here does not define the phase relationship, there is some ambiguity or uncertainty in the description. Phase almost certainly plays some role. Again, presumably 2*Pi of 4-wave vectors could describe the same 4-momentum vector. Once one starts taking waves to be the primary description of a system, the particle aspect gets lost, or smeared out. Once the gradient operation is added to the mix, one gets what is essentially a diffusion equation for waves, in which the particle aspect is lost. Thus, a probabilistic interpretation is needed, showing that the particle is located somewhere/when in the spacetime, but can't quite be pinned down exactly. My bet is that if the phases could be found, the exact locations of particle events would arise.

This remains a work in progress.

Reference papers/books can be found in the 4-Vectors Reference.

Email me, especially if you notice errors or have interesting comments.
Please, send comments to John
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