A physical derivation of Quantum Mechanics (QM) using only the assumptions of Special Relativity (SR) as a starting point...
Quantum Mechanics is not only compatible with Special Relativity, QM is derivable from SR!
The SRQM Interpretation of Quantum Mechanics
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The following is a derivation ofQuantum Mechanics (QM) fromSpecial Relativity (SR).
It basically highlights the few extra physical assumptions necessary to generate QM given SR as the base assumption.
The Axioms of QM are not required, they emerge instead as Principles of QM based on SR derivations.
See also presentation as PDF: SRQM.pdf or as OpenOffice Presentation SRQM.odp
There is also a more basic derivation at SRQM-RoadMap.html
Also, see 4-Vectors & Lorentz Scalars Reference for lots more info on four-vectors (4-vectors) in general
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 Schrödinger equation, at which point we have Quantum Mechanics. In the more advanced chapters, the books will then say that we need a wavefunction that obeys Special Relativity, which the Schrödinger equation does not. They then proceed by positing the Klein-Gordon and Dirac equations, saying that they are the relativistic versions of the Schrödinger equation. It is then shown that these versions of QM agree very nicely with all of the requirements of SR, 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 intrigued me is this:Quantum Mechanics seems to join up very well with Special Relativity, but not with General Relativity (GR). Why?
Thinking along that line led me to the following ideas: Why do the textbooks start with the QM Schrödinger equation, which is known to be non-relativistic, 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 the Schrödinger equation is actually the low-velocity (low-energy) limit of the Klein-Gordon equation, just as Newtonian physics is the low-velocity limit of Relativistic physics. Can you get Quantum Mechanics without the starting point of the standard QM axioms? Can the axioms themselves actually be derived from something that makes a little more sense, that is a little more connected to other known physics? So, starting with SR, and it's two simple axioms (Invariance of the Measurement Interval, Constancy of LightSpeed), 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.
So, let's us proceed from the following assumptions:
GR is essentially correct, and SR is the "flat spacetime" limiting-case of GR.
SR is "more correct" than classical mechanics, in that all classical mechanics is just the low-velocity limiting-case of SR.
QM is not a "separate" theory which just happens to hook up nicely with SR,it may be derivable from SR.
Anything posited as fundamental due to the Schrödinger equation is actually just the low-velocity approximation of the Klein-Gordon equation.
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The short summary goes like this:
Start with GR
SR is "flat spacetime" limiting-case of GR
SR includes the following: Invariance of Interval Measure, Minkowski Spacetime, Poincare Invariance, Description of Physics by Tensors and4-Vectors, LightSpeed Constant (c)
Standard SR 4-Vectors include: 4-Position, 4-Velocity, 4-Momentum,4-WaveVector, 4-Gradient.
Relations between these SR 4-Vectors are found empirically, no QM axioms are necessary - The relations turn out to be Lorentz Invariant Scalars.
These scalars include: Proper Time (τ), Particle Rest Mass (m_{o}),Universal Action Constant (ћ), Imaginary Unit (i).
Only this, and no axioms from QM, are enough to generate:
SR/QM Plane Waves
The Schrödinger RelationP = i ћ ∂
Operator formalism
Unitary Evolution
Non-zero Commutation of Position/Momentum
Relativistic Harmonic Oscillation of Events
The Klein-Gordon Relativistic Wave Equation
The KG Equation implies a QM Superposition Principle because it is a*Linear* Wave PDE.
The Schrödinger Equation is actually just the low-velocity approximation of the Klein-Gordon Equation.
Once you have a Relativistic Wave Equation, you have QM.
However, one can go even further...
The Casimir Invariants of the Poincare Group give Mass *and* Spin - i.e.Spin comes not from QM, but from Poincare Invariance...
Some more exotic SR 4-vectors include: 4-SpinMomentum, 4-VectorPotential, 4-CanonicalMomentum
The 4-SpinMomentum is still a 4-Momentum, but includes the proper machinery for the particle to interact properly spin-wise with an external 4-VectorPotential
The 4-VectorPotential is empirically related to the other 4-Vectors by a charge (q)
At this point, we now have all the stuff (mass, charge, spin, 4-position,4-velocity) needed to describe a particle in spacetime.
Lorentz Invariance of this 4-SpinMomentum gives a Relativistic Pauli equation.
This Relativistic Pauli equation can be shown as the the source of all the usual QM equations: Dirac, Weyl, Maxwell, Pauli, Klein-Gordon, Schrödinger,etc.
And at no point do we need quantum axioms - the principles of QM emerge from this formalism.
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*Note* Numeric subscripts and superscripts on variables inside the vector parentheses typically represent tensor indices, not exponents
In the following, I use the Time-0th-Positive SR metric sign convention η_{μν} = η^{μν} = DiagnolMatrix[+1,-1,-1,-1]
I use this primarily because it reduces the number of minus signs in Lorentz Scalar Magnitudes, since there seem to be more time-like physical 4-vectors than space-like.
Also, this sign convention is the one matched by the QM Schrödinger Relations later on...
I always choose to have the 4-Vector refer to the upper index tensor of the same name. {eg.A = A^{μ}}
In addition, I like the convention of having the (c) factor in the temporal part for correct dimensional units. {eg. 4-Position R = (ct,r)}
This allows the SR 4-Vector name to match the classical 3-vector name,
which is useful when considering Newtonian limiting cases.
I will use UPPER case bold for 4-vectors, and lower case bold for 3-vectors.
All SR 4-Vectors have the following properties: ==================================
A = A^{μ} = (a^{0},a^{i}) = (a^{0}, a) = (a^{0},a^{1},a^{2},a^{3}) = > (a^{t},a^{x},a^{y},a^{z}): A typical 4-vector
where A_{μ} = η_{μν}A^{ν} and A^{μ} = η^{μν}A_{ν}: Tensor index lowering and raising with the Minkowski Metric
A·B = A^{μ} η_{μν} B^{ν} = A_{ν} B^{ν} = A^{μ} B_{μ} = +a^{0}b^{0}-a·b =
+a^{0}b^{0}-a^{1}b^{1}-a^{2}b^{2}-a^{3}b^{3} The Scalar Product relation, used
to make Invariant Lorentz Scalars
If the scalar product is between tensors with multiple indices, then one
should use tensor indices for clarity, otherwise the equation remains
ambiguous.
{eg. U·F^{μν} = U^{α}·F^{μν} = ? = > U^{α}η_{αμ}F^{μν} = U_{μ}F^{μν} or U^{α}η_{αν}F^{μν} = U_{ν}F^{μν} }
Importantly, A·B = (a^{0}_{o}b^{0}_{o}) = A_{o}·B_{o} and A·A = (a^{0}_{o})^{2} = A_{o}·A_{o}, the Lorentz Scalar Product can quite often be set to the "rest values" of the temporal component.
This occurs when the 4-Vector A is Lorentz-Boosted to a frame in which the spatial component is zero: A = (a^{0}, a) == > A_{o} = (a^{0}_{o}, 0)
[A·B > 0] --> Time-Like [A·B = 0] --> Light-Like / Photonic / Null [A·B < 0] --> Space-Like
The Invariant Rest Value of the Temporal Component Rule:
========================================== β = v/c = u/c
4-UnitTemporal T = γ(1,β) = U/c
4-Velocity U = γ(c,u) = cT
Generic 4-Vector A = (a^{0},a) A·T = (a^{0}, a)·γ(1,β) = γ(a^{0}*1 - a·β) = γ(a^{0} - a·β) = (1)(a^{0}_{o} - a·0) = a^{0}_{o} A·T = a^{0}_{o}
The Lorentz Scalar product of any 4-Vector with the 4-UnitTemporal gives the Invariant Rest Value of the Temporal Component.
This makes sense from a vector viewpoint - you are taking the projection
of the generic vector along a unit-length vector in the time direction. A·U = c*a^{0}_{o}
The Lorentz Scalar product of any 4-Vector with the 4-Velocity gives c*Invariant Rest Value of the Temporal Component.
It's the same thing, just multiplied by (c).
I will call these ( A·T = a^{0}_{o} or A·U = c*a^{0}_{o} ) the "Invariant Rest Value of the Temporal Component Rule".
This will get used extensively later on...
There is an analogous relation with the 4-UnitSpatial.
The Scalar Product-Gradient-Position Relation:
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4-Position R = (ct,r)
4-Gradient ∂ = ∂_{X} = (∂_{t}/c,-∇)
Generic 4-Vector A = (a^{0},a), which is not a function of R
A·R = (a^{0},a)·(ct,r) = (a^{0}*ct - a·r) = Θ is equivalent to ∂[Θ] = ∂[A·R] = A
*Note* f = f(t,x) ==> df = (∂_{t}f) dt + (∂_{x}f) dx f = ∫df = ∫(∂_{t}f) dt + ∫(∂_{x}f) dx f ==> (∂_{t}f) ∫dt + (∂_{x}f) ∫dx = ∂_{t}f *t + ∂_{x}f *x {if the partials are constants wrt. t and x, which was the condition from A not a function of R}
This comes up in the SR Phase and SR Analytic Mechanics.
Basis Representation & Independence (Manifest Covariance):
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When the components of the 4-vector { A }are in (time scalar,space 3-vector) form { (a^{0},a) }then the 4-vector is in spatial basis invariant form.
Once you specify the spatial components individually, you have picked a
basis or representation. I indicate this by using { = > }.
e.g. 4-Position X = (ct,x) = {Space Basis independent representation}
= > (ct,x,y,z) = {Cartesian/rectangular representation}
= > (ct,r,θ,z) = {Cylindrical representation}
= > (ct,r,θ,φ) = {Spherical representation}
These can all indicate the same 4-Vector, but the components of the 4-vector will vary in the different bases.
Now, once you are in a space basis invariant form, e.g. X = (ct,x), you can still do a Lorentz boost and still have the same 4-Vector X. It is only when using 4-Vectors directly, (eg. X·Y, X+Y), that you have full Spacetime Basis Independence. Knowing
this, we try to find as many relations as possible in 4-Vector and
Tensor format, as these are applicable to all observers.
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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-vectorsthat 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 events, at which there may or may not be 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 occurs purely within SR - ex. Relativistic Optics,Relativistic Doppler Effect.
*Fields occur purely due to SR Potential Momentum.
*QM is generated simply by allowing the particles to have imaginary/complex components within spacetime.
*The Quantum Superposition Principle, usually assumed as axiomatic, is a consequence of the Klein-Gordon equation being a linear wave PDE.
Notation and Properties of SR 4-Vectors
g_{μν} = g^{μν} = > η^{μν} = DiagnolMatrix[1,-1,-1,-1] Minkowski Spacetime Metric: This is the "flat"spacetime of SR
All SR 4-Vectors have the following properties: A = A^{μ} = (a^{t},a^{x},a^{y},a^{z}) = (a^{0},a^{1},a^{2},a^{3}) = (a^{0},a)A typical 4-vector
A_{μ} = (a_{t},a_{x},a_{y},a_{z}) = (a_{0},a_{1},a_{2},a_{3}) = (a^{t},-a^{x},-a^{y},-a^{z}) = (a^{0},-a^{1},-a^{2},-a^{3}) = (a_{0},-a)A typical 4-covector; we can always get the 4-vector form with A^{μ} = η^{μν}A_{ν} A·B = η_{uv }A^{u }B^{ν} = A_{ν}B^{ν} = A^{μ} B_{μ}^{ } = +a^{0}b^{0}-a^{1}b^{1}-a^{2}b^{2}-a^{3}b^{3} = +a^{0}b^{0}-a·b The Scalar Product relation,used to make Invariant Lorentz Scalars
[A·B > 0] --> Time-Like [A·B = 0] --> Light-Like / Null [A·B < 0] --> Space-Like
Useful Quantities
γ[v] = 1 / √[1-(v/c)^{2}] : Lorentz Scaling Factor (gamma factor)
τ[v,t] = t / γ : Proper Time
Sqrt[1+x] = √[1+x] ~ (1+x/2) for |x|<<1 : Math relation often used to simplify Relativistic eqns. to Newtonian eqns.
(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) = γm_{o}(c,u)
4-CurrentDensity J = (cρ,j) = γρ_{o}(c,u)
4-WaveVector K = (ω/c,k) = (ω/c,nω/v_{phase}) = (ω/c)(1,β) = (1/cT,n/λ)
4-Gradient ∂ = (∂_{t}/c,-∇) = > (∂/c∂t,-∂/∂x,-∂/∂y,-∂/∂z) = (∂_{t}/c,-∂_{x},-∂_{y},-∂_{z})
4-VectorPotential A = (φ/c,a)
4-PotentialMomentum Q = qA = q(φ/c,a) = (U/c,p)*includes effect of charge q*
4-TotalMomentum P_{T} = (H/c,p_{T}) = P +Q = P + qA
4-TotalGradient D = ∂ + iq/ћA = 4-Gradient∂ + effects of Vector Potential
Fundamental/Universal Relations
R = R U = dR/dτ "4-Velocity is the derivative of 4-Position wrt. proper time" P = m_{o}U J = ρ_{o}U K = P / ћ ∂ = -iK Q = qA P_{T} = (H/c,p_{T}) = P +Q D = ∂ + iq/ћA "whereA is the (EM) vector potential and q is the (EM) charge"
Derived Physical Constants (Scalar Products of Lorentz Vectors give Lorentz Scalars)
Start with a special relativistic spacetime for which the invariant measurement interval is given byR·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.
In this convention, 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:
All of the Special Relativistic notation applies to the concept of events.
Events are simply points in spacetime. The measurement interval between points is an invariant.
Now, let's examine the interesting events...
There exist particles (which can carry information) that move about in this spacetime.
Each particle is located 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 forward in time.
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 be able to move around within the spacetime.
The 4-Velocity of an event is given by U = 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 / √[1-(v/c)^{2}].
This particle, if its rest mass m_{o}>0, moves only in the direction of +time along its own worldlineU_{worldline} = (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 m_{o} = 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:U_{light-like} = Infinite c(1,n), wheren 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 velocityu with respect to another particle or an observer.
We have the relation √(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.
One of the basic properties of particles is that of mass. Each particle has a rest mass m_{o}.
Rest mass is simply the mass as measured in a frame at rest with respect to an observer.
This mass, along with the velocity of a particle, gives 4-Momentum P = m_{o}U.
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 relationP = m_{o}U gives 4-Momentum P = (E/c,p) = m_{o}U = γm_{o}(c,u).
This gives the Einstein Mass-Energy relation, E = γm_{o}c^{2},or E = mc^{2} where m = (γm_{o}).
Note that for light-like particles, the result using this formula is undefined since E_{light-like} = Infinite 0 c^{2}.
Presumably, the m = (γm_{o}) factor must scale in some way(i.e. like a delta function) to give reasonable results.
Also, there is a lot of confusion over whether m is the actual mass or not.
A simple thought experiment clears this up. Imagine an atom at rest,having rest mass m_{o}.
Now imagine an observer moving past the atom at near light speed.
The apparent mass of the atom to the moving observer is m = (γm_{o}).
Now imagine this observer accelerating to ever greater speeds.
The atom is sitting happy and unchanging in its own rest frame.
However, once the observer is going fast enough, this apparent mass m = (γm_{o}) could be made to exceed that necessary to create a black hole.
As that would be an irreversible event, the gamma factor γ must simply be a measure of the relative velocities of the two events.
So, the true measure of actual mass is just the rest mass m_{o}.
Waves/Null-Particles:
The energy of null/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 thatK = P / ћ = (ω/c,k) = 1/ћ (E/c,p).
We should note here that h (ћ = h/2π) is an empirical constant, which can be measured with no assumptions about QM, just as c is an empirical constant which can be measured with no assumptions about SR.
Planck discovered h based on statistical-mechanics/thermodynamic considerations of the black-body problem.
Einstein applied Planck's idea to photons in the photoelectric effect to give E = ћ ω and the idea of photons as particle quanta.
de Broglie realized that every particle, massive or massless, has 3-vector momentum p = ћk.
Putting it all together naturally produces 4-vector P = ћK = (E/c,p) = ћ(ω/c,k).
Note also that the 4-WaveVector (a wave-like object) is just a constant, ћ,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 = γm_{o}c^{2} = ћ ω ), and also gives the relation for m = (γm_{o}) = ω ћ/c^{2} = (γω_{o})ћ/c^{2}.
Note that massive particle would have rest frequency ω_{o} , which would look like (γ ω_{o}) to an observer, while massless particles simply have frequency ω.
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-WaveVector is everywhere in the direction of propagation of the wave surfaces. From this structure, one obtains relativistic/wave optics, without ever mentioning QM.
I believe that there is more to the 4-WaveVector than other people have figured on (i.e. more importance to the overall phase Φ 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 cyclic period, without the need for a medium to be waving in. Also, note that the phase of the 4-WaveVector was not defined. Presumably, (2π) of 4-WaveVec's could have the same 4-vector K . 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∂ = ∂^{μ} = (∂_{t}/c,-∇) = (∂_{t}/c,-del) = > (∂/c∂t,-∂/∂x,-∂/∂y,-∂/∂z) = (∂_{t}/c,-∂_{x},-∂_{y},-∂_{z}) , 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, for the upper tensor index, unlike the other "physical type" vectors.
∂·X = ∂/c∂t[ct]+∇·x = ∂t/∂t+∇·x = 4.
This tells us the number of spacetime dimensions.
When it is applied to the 4-CurrentDensity, it leads to the Conservation of Charge equation. ∂·J = ∂/c∂t[cρ]+∇·j = ∂ρ/∂t+∇·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]+∇·p = (1/c^{2})∂E/∂t +∇·p = 0. ∂E/∂t+c^{2}∇·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.
Additionally,U·∂ = γ(∂/∂t +u·∇) = γ d/dt = d/dτ
Showing that the derivative w.r.t. Proper Time is a Lorentz Scalar Invariant.
The 4-Gradient∂ = (∂_{t}/c,-∇) = (∂_{t}/c,-del) is an SR functional that gives the structure of Minkowski Spacetime.
The Lorentz Scalar Product∂·∂ = (∂_{t}/c,-∇)·(∂_{t}/c,-∇) = (∂_{t}/c)^{2} -∇·∇
gives the d'Alembertian equation / wave equation. The d'Alembert operator is the Laplace operator of Minkowski Space. Despite being a functional, the d'Alembertian is still a Lorentz Scalar Invariant.
The Green's function G[X-X'] for the d'Alembertian is defined as (∂·∂)G[X-X'] = δ^{4}[X-X']
So, given all the above, we have clearly shown that∂ is SR4-vector, not something from QM.
Now, let's perform some pure SR math based on our SR 4-vector knowledge
Now, let's make a SR function f
let f = ae^b(K∙X), which is just a simple exponential function of 4-vectors
then∂[f] = (bK)ae^b(K∙X) = (bK)f
and∂∙∂[f] = b^{2}(K∙K)f = (bω_{o}/c)^{2}f
Note that { b = -i } is an interesting choice – it leads to SR Plane Waves,which we observe empirically, e.g. EM Plane Waves...
This gives: ∂[f] = (-iK)ae^-i(K∙X) = (-iK)f ∂[f] = (-iK)f ∂ = -iK
Now comes Quantum Mechanics (QM)!
Now then, based on empirical evidence:
QM (and enhancements like QED and QFT) have given the correct calculation/approximation of more phenomena than any other theory, ever.
We have the following simple relation:∂ = -iK orK = i∂.
This innocent-looking, very simple relation gives all of Standard QM.
It does this in a number of ways, one of which is by providing the Schrödinger relationP = ћK = i ћ ∂.
In component form this is (E = i ћ ∂/∂t) and (p = -iћ∇).
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 Operator Formalism, Unitary Evolution, and Wave Structure Axioms of QM, which governs how the state of a quantum system evolves in time.
We have:
[∂] = -iK:Operator Formalism ∂ = [ -i ]K:Unitary Evolution ∂ = -i [K] :Wave Structure
One also finds that SR events oscillate with a rest freq that is proportional to rest mass. U·∂ = γ(∂/∂t +u·∇) = γ d/dt = d/dτ
d/dτ = U·∂
d/dτ = U·(-iK)
d/dτ = U·(-i/ћP)
d/dτ = U·(-im_{o}/ћ U)
d/dτ = (-im_{o}/ћ )U·U
d/dτ = (-im_{o}c^{2}/ћ)
d/dτ = (-iω_{o}m_{o}c^{2}/ћω_{o})
d/dτ = (-iω_{o})
d^{2}/dτ^{2} = -(ω_{o})^{2}
Now, apply this to the 4-Position...
d^{2}X/dτ^{2} = -(ω_{o})^{2}X
This is the differential equation of a relativistic harmonic oscillator!
Quantum events oscillate at their rest-frequency.
Likewise for the momenta:
d^{2}P/dτ^{2} = -(ω_{o})^{2}P
Next, let's look at Quantum Commutation Relations...4-PositionX = (ct,x)4-Gradient∂ = (∂t/c,-∇)Then, purely from math... ================== Let ψ be an arbitrary function.X[ψ] = Xψ,∂[ψ] = ∂[ψ]X[∂[ψ]] = X∂[ψ]∂[X[ψ]] = ∂[Xψ] = ∂[X]ψ +X∂[ψ] ∂[Xψ]-X∂[ψ] = ∂[X]ψ
now with commutator notation
[∂,X]ψ = ∂[X]ψ
And since ψ was an arbitrary function...
[∂,X] = ∂[X]
[∂,X] = ∂[X] = (∂t/c,-∇)[(ct,r)] = (∂_{t}/c,-∂_{x},-∂_{y},-∂_{z})[(ct,x,y,z)] = Diag[1,-1,-1,-1] = η^{μν} = Minkowski Metric
[∂,X] = η^{μν} = Minkowski Metric
================== At this point,we have established purely mathematically, that there is a non-zero commutation relation between the SR 4-Gradient and SR 4-PositionThen, from our empirical measurements...we find that ∂ = - i K so
[∂,X] = η^{μν}[-iK,X] = η^{μν}
- i [K,X] = η^{μν}
[K,X] = i η^{μν}
Then, from our empirical measurements...we know that K = (1/ћ)P
[K,X] = i η^{μν}[(1/ћ)P,X] = i η^{μν}
(1/ћ)[P,X] = i η^{μν}
[P,X] = i ћη^{μν}
[X^{μ},P^{ν}] = - i ћη^{μν}and, looking at just the spatial part
[x_{i},p_{j}] = i ћ δ_{ij}Hence, we have derived the standard QM commutator rather than assume it as an axiom...
Let's summarize a bit:
We used the following relations:(particle/location-->movement/velocity-->mass/momentum-->wave duality-->spacetime structure)
With the exception of 4-Velocity being the derivative of 4-Position, all of these relations are just constants times other 4-Vectors.
R = (ct,r)
particle/location
U = dR/dτ
movement/velocity
P = m_{o}U
mass/momentum
K = 1/ћP
wave duality
∂ = -iK
spacetime structure
By applying the Scalar Product law to these relations, we get: U·U = (c)^{2} P·P = (m_{o}c)^{2} K·K = (m_{o}c/ћ)^{2} ∂·∂ = (-im_{o}c/ћ)^{2} = -(m_{o}c/ћ)^{2}
Let's look at that last equation. ∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂^{2}/c^{2}∂t^{2}-∇·∇ = -(m_{o}c/ћ)^{2},
∂^{2}/c^{2}∂t^{2} = ∇·∇-(m_{o}c/ћ)^{2}
This is the basic, free-particle, Klein-Gordon equation, the relativistic cousin of the Schrödinger equation!
It is the relativistically-correct, quantum wave-equation for spinless (spin 0) particles.
We have apparently discovered QM by 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 (a linear PDE) obeys the 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 simply reinterpretation of the continuity equation,∂·J = ∂/c∂t(cp) +∇·j = ∂p/∂t +∇·j = 0, whereJ 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 usingD = ∂ + iq/ћA whereA is the EM vector potential and q is the EM charge,
and allowingD·D = -(m_{o}c/ћ)^{2} to be the more correct EM quantum wave equation.
Now, let's back up a bit toP·P = (m_{o}c)^{2} P·P - (m_{o}c)^{2} = 0
(E/c)^{2} -p·p - (m_{o}c)^{2} = 0
E^{2} - c^{2}p·p - (m_{o}c^{2})^{2} = 0
this can be factored into...
[ E - cα·p - β(m_{o}c^{2}) ] [ E + cα·p+ β(m_{o}c^{2}) ] = 0
where:
E andp are quantum operators, α and β are matrices which must obey α_{i}β = -βα_{i},α_{i}α_{j} = -α_{j}α_{i}, α_{i}^{2} = β^{2} = I
The left hand term can be set to 0 by itself, giving...
[ E - cα·p - β(m_{o}c^{2}) ] = 0, which is the Dirac equation, which is correct for spin 1/2 particles
Potentials/Fields:
Let's back up to the 4-Momentum equation. Momentum is not just a property of individual particles, but also of fields.
These fields can be described by 4-vectors as well.
One such relativistically invariant field is the 4-VectorPotential A,which is itself a function of 4-Position X.
Typically, we deal with the Electromagnetic (EM) 4-VectorPotential, but it could be any kind of relativistic charge potential...
4-VectorPotential A[X] = A[(ct,x)] = (φ/c,a) = (φ[(ct,x)]/c, a[(ct,x)]), where the [(ct,x)] means is a function of time t and position x.
While a particle exists as a worldline over spacetime, the 4-VectorPotential exists over all spacetime.
The 4-VectorPotential can carry energy and momentum, and interact with particles via their charge q.
PotentialMomentum:
One may obtain the PotentialMomentum 4-vector by multiplying by a charge q,Q = qA
4-PotentialMomentum Q = qA = q(φ/c,a) = (U/c,p)
The 4-TotalMomentum is then given by P_{T} = P +Q
This includes the momentum of particle and field, and it is the locally conserved quantity.
4-TotalMomentum P_{T} = (H/c,p_{T}),where these are the TotalEnergy = Hamiltonian and 3-TotalMomentum. P = P_{T} -Q = m_{o}U
Now working back, we can make our dynamic 4-Momentum more generally,including the effects of potentials.
4-Momentum P = (E/c,p) = (H/c - U/c,p_{T}-p_{EM}) = (H/c - qφ/c,p_{T} - qa)
The dynamic 4-momentum of a particle thus now has a component due to the 4-VectorPotential,
and reverts back to the usual definition of 4-momentum in the case of zero 4-VectorPotential.
Likewise, following the same path as before... K = P / ћ
4-WaveVector K = (ω_{T}/c -(q/ћ)φ/c,k_{T} - (q/ћ)a) ∂ = -iK
4-Gradient ∂ = (∂_{T}/c∂t- (iq/ћ)φ/c,-∇_{T} - (iq/ћ)a) = (∂_{t}/c,-∇)
Define 4-TotalGradient D = ∂ +iq/ћA This is the concept of "Minimal Coupling"
Minimal Coupling can be extended all the way to non-Abelian gauge theories and can be used to write down all the interactions of the Standard Model of elementary particles physics between spin-1/2"matter particles" and spin-1 "force particles"
Minimal Coupling applied to the Dirac Eqn. leads to the Spin Magnetic Moment-External Magnetic Field coupling W = -γ_{e}S·B, where γ_{e} = q_{e}/m_{e},the gyromagnetic ratio.
The corrections to the anomalous magnetic moment come from minimal coupling applied to QED
In addition, we can go back to the velocity formula:
u = c^{2} (p)/(E) = c^{2} (p_{T} - qa)/(H - qφ)
Lagrangian/Hamiltonian Formalisms:
The whole Lagrangian/Hamiltonian connection is given by the relativistic identity:
( γ - 1/γ ) = ( γβ^{2} )
Now multiply by your favorite Lorentz Scalars... In this case for a free relativistic particle
( γ - 1/γ )(P·U) = ( γβ^{2} )(P·U)
( γ - 1/γ )(m_{o}c^{2}) = ( γβ^{2} )(m_{o}c^{2})
( γm_{o}c^{2} - m_{o}c^{2}/γ ) = γm_{o}c^{2}β^{2}
( γm_{o}c^{2} - m_{o}c^{2}/γ ) = γm_{o}v^{2}
( γm_{o}c^{2} ) + (- m_{o}c^{2}/γ ) = γm_{o}u·u
( γm_{o}c^{2} ) + (- m_{o}c^{2}/γ ) = (p·u )
( H ) + ( L) = (p·u )
The Hamiltonian/Lagrangian connection falls right out
Now, including the effects of the 4- Vector Potential A = (φ/c,a){ = (φ_{EM}/c,a_{EM}) for EM potential }
Momentum due to Potential Q = qA Total Momentum of system P_{T} = Π = P +Q = P + qA = m_{o}U + qA = (H/c,p_{T}) = (γm_{o}c+q φ/c,γm_{o}u+qa_{)}
A·U = γ(φ -a·u ) = φ_{o
}P·U = γ(E -p·u ) = E_{o
}P_{T}·U = E_{o}+ qφ_{o} = m_{o}c^{2}+ qφ_{o}
I assume the following: A = (φ_{o}/c^{2})U = (φ/c,a) = φ_{o}/c^{2} γ(c,u) = ( γφ_{o}/c,γφ_{o}/c^{2}u) giving (φ = γφ_{o} anda = γφ_{o}/c^{2}u)
This is analogous to P = E_{o}/c^{2}U
L = -(P_{T}·U)/γ
L = -((P +Q)·U)/γ
L = -(P·U +Q·U)/γ
L = -P·U/γ -Q·U/γ
L = -m_{o}U·U/γ - qA·U/γ
L = -m_{o}c^{2}/γ - qA·U/γ
L = -m_{o}c^{2}/γ - q(φ/c,a)·γ(c,u)/γ
L = -m_{o}c^{2}/γ - q(φ/c_{,}a)·(c,u)
L = -m_{o}c^{2}/γ - q(φ -a·u)
L = -m_{o}c^{2}/γ - qφ + qa·u
L = -m_{o}c^{2}/γ - qφ_{o}/γ
L = -(m_{o}c^{2} + qφ_{o})/γ
H = γ(P_{T}·U)
H = γ((P +Q)·U)
H = γ(P·U +Q·U)
H = γP·U + γQ·U
H = γm_{o}U·U + γqA·U
H = γm_{o}c^{2} + qγφ_{o}
H = γm_{o}c^{2} + qφ assumingA = (φ_{o}/c^{2})U
H = ( γβ^{2} + 1/γ )m_{o}c^{2} + qφ
H = ( γm_{o}β^{2}c^{2} + m_{o}c^{2}/γ) + qφ
H = ( γm_{o}v^{2} + m_{o}c^{2}/γ) + qφ
H = p·u + m_{o}c^{2}/γ + qφ
H = E + qφ
H = ± c√[m_{o}^{2}c^{2}+p^{2}]+ qφ
H = ± c√[m_{o}^{2}c^{2}+(p_{T}-qa)^{2}]+ qφ
Let's now show that the Schrödinger equation is just the low energy limit of the Klein-Gordon equation.
We now let the Klein-Gordon equation use the Total Gradient, so now our wave equation uses EM potentials. D·D = -(m_{o}c/ћ)^{2}(∂ + iq/ћA)·(∂+ iq/ћA) + (m_{o}c/ћ)^{2} = 0
letA' = (iq/ћ)A
let M = (m_{o}c/ћ)_{
}then (∂ +A')·(∂ +A') + (M)^{2} = 0
∂·∂ +∂·A' + 2A'·∂ +A'·A' + (M)^{2} = 0
now the trick is that factor of 2, it comes about by keeping track of tensor notation...
a weakness of strict 4-vector notation
let the 4-Vector potential be a conservative field, then∂·A = 0
(∂·∂) + 2(A'·∂) + (A'·A') +(M)^{2} = 0
take the negative root
(i ћ)( ∂_{t} ) ~ (qφ ) + [(m_{o}c^{2})- (ћ)^{2}(∇ + iq/ћa)^{2}/2m_{o} ]
(i ћ)( ∂_{t} ) ~ (qφ ) + (m_{o}c^{2})- (ћ)^{2}(∇ + iq/ћa)^{2}/2m_{o}
call (qφ ) + (m_{o}c^{2}) = V[x]
(i ћ)( ∂_{t} ) ~ V[x] - (ћ)^{2}(∇ +iq/ћa )^{2}/2m_{o
}
typically the vector potential is zero in most non-relativistic settings
(i ћ)( ∂_{t} ) ~ V[x] - (ћ)^{2}(∇)^{2}/2m_{o}
And there you have it, the Schrödinger Equation with a potential
The assumptions for non-relativistic equation were:
Conservative field A, then∂·A = 0
(∇ +a' )^{2}/ i^{2}(M)^{2} = (∇ +a' )^{2}/ i^{2}(m_{o}c/ћ)^{2} = (ћ)^{2}(∇ +a' )^{2}/i^{2}(m_{o}c)^{2} is near zero
i.e. (ћ)^{2}(∇ +a')^{2} << (m_{o}c)^{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∂·∂ = -(m_{o}c/ћ)^{2}is shown to be a limiting case for A_{EM} = 0.
Again, see the 4-Vectors Reference for more on this.
Now, let's examine something interesting... ∂·∂ = -(m_{o}c / ћ)^{2}: Klein-Gordon Relativistic Wave eqn. ∂ = -i/ћP ∂·(-i/ћP) = -(m_{o}c/ ћ)^{2} ∂·(P) = - i (m_{o}c)^{2}/ ћ ∂·(P) = 0 - i (m_{o}c)^{2}/ ћ
but,∂·(P) = Re[∂·(P)], by definition, since the4-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 toQM-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 = m_{o}U
Momentum is just the Rest Mass of the particle times its velocity
K = P /ћ
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...
D = ∂ + (iq/ћ)A
The particle with minimal coupling interaction in a potential field
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. [-∇ |V> = -ik |V>]
A solution to this equation is:
|V> = v_{n} e^(-iK_{n}·R) |V_{n}> where v_{n} is a real number, |V_{n}> is an eigenstate(stationary state)
Generally, |V> can be a superposition of eigenstates |V_{n}>
N
|V> = Sum [v_{n} e^(-iK_{n}·R) |V_{n}>]
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.