New X'^{μ}  =  4Tensor Antisymmetric Lorentz Transform M^{μν} Rotations j = M^{ab} Boosts k = M^{0b} = M^{b0} 3 + 3 = 6 
Original X^{μ}  +  4Vector SpaceTime Translation ΔX^{μ} ~ P^{μ} Time Translation H = P^{0} Space Translation p = P^{i} 1 + 3 = 4  
 = 

 + 
 
Total of 6 + 4 = 10 parameters Poincaré Transform = Lorentz Transform + SpaceTime Translation X'^{μ} = M^{μ}_{ν} X^{ν} + ΔX^{μ} where colors indicate ^{ }

Translational Operator 
∂^{μ}  K^{μ}  P^{μ} 
Equivalent  = ∂^{μ}  = i∂^{μ}  = iћ∂^{μ} 
Normal Commutator 
[∂^{μ}, X^{ν}] = η^{μν}  [K^{μ},X^{ν}] = iη^{μν}  [P^{μ}, X^{ν}] = iћη^{μν} 
Reversed Commutator 
[X^{ν}, ∂^{μ}] = η^{μν}  [X^{ν}, K^{μ}] = iη^{μν}  [X^{ν}, P^{μ}] = iћη^{μν} 
Rotational Momentum Operator M 
M^{μν} 
M^{μν} 
M^{μν}  Dimensionless Rotational Operator O 
O^{μν}  O^{μν}  O^{μν} 
Equivalent  = iћ(X^{μ}∂^{ν}  X^{ν}∂^{μ})  = ћ(X^{μ}K^{ν}  X^{ν}K^{μ})  def. = (X^{μ}P^{ν}  X^{ν}P^{μ}) 
Equivalent  def. = (X^{μ}∂^{ν}  X^{ν}∂^{μ}) 
= (1/i)(X^{μ}K^{ν}  X^{ν}K^{μ})  = (1/iћ)(X^{μ}P^{ν}  X^{ν}P^{μ}) 
Normal Commutator 
[M^{μν}, ∂^{ρ}] = iћ(η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) 
[M^{μν}, K^{ρ}] = iћ(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) 
[M^{μν}, P^{ρ}] = iћ(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) 
Normal Commutator 
[O^{μν}, ∂^{ρ}] = (η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) 
[O^{μν}, K^{ρ}] = (1/i)(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) 
[O^{μν}, P^{ρ}] = (1/iћ)(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) 
Reversed Commutator 
[∂^{ρ}, M^{μν}] = iћ(η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) = iћ(η^{ρμ}∂^{ν}  η^{ρν}∂^{μ}) 
[K^{ρ}, M^{μν}] = iћ(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) = iћ(η^{ρμ}K^{ν}  η^{ρν}K^{μ}) 
[P^{ρ}, M^{μν}] = iћ(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) = iћ(η^{ρμ}P^{ν}  η^{ρν}P^{μ})  Reversed Commutator 
[∂^{ρ}, O^{μν}] = (η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) = (η^{ρμ}∂^{ν}  η^{ρν}∂^{μ})  [K^{ρ}, O^{μν}] = (1/i)(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) = (1/i)(η^{ρμ}K^{ν}  η^{ρν}K^{μ})  [P^{ρ}, O^{μν}] = (1/iћ)(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) = (1/iћ)(η^{ρμ}P^{ν}  η^{ρν}P^{μ}) 
Standard Model Elementary Particle 
Relativistic Wave Equations (RWE)  Relativistic Wave Equations (RWE)  Newtonian Limit ( v << c )  
Particle Type  Spin  Statistics  Field  RQM Massless (m_{o} = 0)  RQM Massive (m_{o} > 0)  QM Massive (m_{o} > 0) 
Fundamental  0  Boson  Lorentz Scalar ψ 
Scalar Wave (∂·∂)ψ = 0 
KleinGordon Equation (∂·∂ + (m_{o}c/ћ)^{2})ψ = 0 
Schrödinger Equation (iħ∂_{t})ψ ~ [(m_{o}c^{2})  (ħ∇)^{2}/2m_{o}]ψ 
Fundamental  1/2  Fermion  Spinor Ψ 
Weyl Equation [(iγ^{μ}∂_{μ})]Ψ = 0 → [(σ^{μ}∂_{μ})]Ψ = 0 
Dirac Equation, Majorana Equation [(iγ^{μ}∂_{μ})  (m_{o}c/ћ)]Ψ = 0 (Γ^{μ}P_{μ})Ψ = (m_{o}c)Ψ iћ(Γ^{μ}∂_{μ})Ψ = (m_{o}c)Ψ 
Pauli Equation (iħ∂_{t})Ψ ~ [(m_{o}c^{2}) + (σ·p)^{2}/2m_{o}]Ψ 
Fundamental  1  Boson  4Vector A 
Maxwell Equation (∂·∂)A = 0 
Proca Equation (∂·∂ + (m_{o}c/ћ)^{2})A = 0 
? 
Composites  3/2  Fermion  SpinorVector  Majorana RaritaSchwinger  RaritaSchwinger Equation  
??  2  Boson  (2,0)Tensor  Graviton?? 
Standard Model Elementary Particle 
Relativistic Wave Equations (RWE)  Relativistic Wave Equations (RWE)  Newtonian Limit ( v << c )  
Particle Type  Spin  Statistics  Field  RQM Massless (m_{o} = 0)  RQM Massive (m_{o} > 0)  QM Massive (m_{o} > 0) 
Fundamental  0  Boson  Lorentz Scalar ψ 
Scalar Wave (D·D)ψ = 0 
KleinGordon Equation (D·D + (m_{o}c/ћ)^{2})ψ = 0 
Schrödinger Equation (iħ∂_{tT})ψ ~ [qφ + (m_{o}c^{2}) + (iħ∇_{T} qa)^{2}/2m_{o}]ψ (iħ∂_{tT})ψ ~ [V + (iħ∇_{T} qa)^{2}/2m_{o}]ψ : with [V = qφ + (m_{o}c^{2})] 
Fundamental  1/2  Fermion  Spinor Ψ 
Weyl Equation ? 
Dirac Equation, Majorana Equation Γ^{μ}(P_{μ}qA_{μ})Ψ = (m_{o}c)Ψ Γ^{μ}(iћ∂_{μ}qA_{μ})Ψ = (m_{o}c)Ψ  Pauli Equation (iħ∂_{tT})Ψ ~ [qφ + (m_{o}c^{2}) + [σ·(p_{T} qa)]^{2}/(2m_{o})]Ψ (iħ∂_{tT})Ψ ~ [qφ + (m_{o}c^{2}) + ([(p_{T} qa)]^{2}  ћq[σ·B])/(2m_{o})]Ψ 
Fundamental  1  Boson  4Vector A 
Maxwell Equation (∂·∂)A = 0 (∂·∂)A^{ν} = μ_{o}J^{ν}: Classical source (∂·∂)A^{ν} = q(ψ̅ γ^{ν} ψ): QED source 
Proca Equation  ? 
Composites  3/2  Fermion  SpinorVector  Majorana RaritaSchwinger  RaritaSchwinger Equation  
??  2  Boson  (2,0)Tensor  Graviton?? 
Written" on the papers Einstein is holding: =================================== R_{μν}  (1/2)g_{μν}R = κT_{μν} (the theory of GR) eV = hν  A (the PhotoElectric Effect) E = mc^{2} (the Equivalence of Energy and Matter) =================================== 
Correlates to: ========== GR QM SR ========== 
V^{0}  V^{1 }  V^{2 }  V^{3 } 
temporal part = V^{0} 
spatial part = V^{i} 
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 3space with an extra, reversedsign dimension, time, added to it.
This interval is Lorentz Invariant.
In this convention, spacelike intervals are ()negative, timelike intervals are (+)positive, and lightlike intervals are (0)null.
One can say that the universe is the set of all possible events in spacetime.
Let's summarize a bit:
We used the following relations:(particle/location→movement/velocity→mass/momentum→wave duality→spacetime structure)
With the exception of 4Velocity being the derivative of 4Position, all of these relations are just constants times other 4Vectors.
R = (ct,r)  particle/location 
U = dR/dτ  movement/velocity 
P = m_{o}U  mass/momentum 
K = 1/ћP  wave duality 
∂ = iK  spacetime structure 
∂^{2}/c^{2}∂t^{2} = ∇·∇(m_{o}c/ћ)^{2} 
This is the basic, freeparticle, KleinGordon equation, the relativistic cousin of the Schrödinger equation!
It is the relativisticallycorrect, quantum waveequation 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, where J is taken to be a "particle"current density.
Relativistic Hamiltonian H = γ(P_{T}·U) 
Relativistic Lagrangain L = (P_{T}·U)/γ 
p_{T}·u = ( γβ^{2})(P_{T}·U) = H + L = γ(P_{T}·U) + (P_{T}·U)/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 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}u^{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 H = ±m_{o}c^{2}√[1 + (p_{T} qa)^{2}/(m_{o}^{2}c^{2})] + qφ 
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 + L = γ(P_{T}·U)  (P_{T}·U)/γ H + L = (γ  1/γ)(P_{T}·U) H + L = ( γβ^{2})(P_{T}·U) H + L = ( γβ^{2})((P + Q)·U) H + L = ( γβ^{2})(P·U + Q·U) H + L = ( γβ^{2})(m_{o}c^{2} + qφ_{o}) H + L = (γm_{o}β^{2}c^{2} + qγφ_{o}β^{2}) H + L = (γm_{o}u·uc^{2}/c^{2} + qφ_{o}γu·u/c^{2}) H + L = (γm_{o}u·u + qa·u) H + L = (p·u + qa·u) H + L = p_{T}·u 
Rest Hamiltonian H_{o} = (P_{T}·U) = H/γ 
Rest Lagrangain L_{o} = (P_{T}·U) = γL 
H_{o} + L_{o} = 0 
Let's now show that the Schrödinger equation is just the low energy limit of the KleinGordon equation.
We now let the KleinGordon 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 4Vector notation
let the 4Vector potential be a conservative field, then∂·A = 0
(∂·∂) + 2(A'·∂) + (A'·A') +(M)^{2} = 0
expanding to temporal/spatial components...
( ∂_{t}^{2}/c^{2}∇·∇ ) +2(φ'/c ∂_{t}/c a'·∇ ) + ( φ'^{2}/c^{2}a'·a') + (M)^{2} = 0
gathering like components
( ∂_{t}^{2}/c^{2} + 2φ'/c ∂_{t}/c+ φ'^{2}/c^{2} )  (∇·∇ +2a'·∇ + a'·a' ) + (M)^{2} = 0
( ∂_{t}^{2} + 2φ'∂_{t} + φ'^{2} ) c^{2}(∇·∇ + 2a'·∇ + a'·a') + c^{2}(M)^{2} = 0
( ∂_{t} + φ' )^{2}  c^{2}(∇ +a')^{2} + c^{2}(M)^{2} = 0
multiply everything by (i ћ)^{2}
(i ћ)^{2}( ∂_{t} + φ' )^{2}  c^{2}(i ћ)^{2}(∇+a' )^{2} + c^{2}(iћ)^{2}(M)^{2} = 0
put into suggestive form
(i ћ)^{2}( ∂_{t} + φ' )^{2} =  c^{2}(iћ)^{2}(M)^{2} + c^{2}(i ћ)^{2}(∇+a' )^{2}
(i ћ)^{2}( ∂_{t} + φ' )^{2} = i^{2}c^{2}(iћ)^{2}(M)^{2} + c^{2}(i ћ)^{2}(∇+a' )^{2}
(i ћ)^{2}( ∂_{t} + φ' )^{2} = i^{2}c^{2}(iћ)^{2}(M)^{2} [1 + c^{2}(i ћ)^{2}(∇+a' )^{2}/ i^{2}c^{2}(iћ)^{2}(M)^{2} ]
(i ћ)^{2}( ∂_{t} + φ' )^{2} = i^{2}c^{2}(iћ)^{2}(M)^{2} [1 + (∇ +a')^{2}/ i^{2}(M)^{2} ]
take Sqrt of both sides
(i ћ)( ∂_{t} + φ' ) = ic(i ћ)(M) Sqrt[1 + (∇ +a' )^{2}/ i^{2}(M)^{2}]
use Newtonian approx Sqrt[1+x] ~ ±[1+x/2] for x<<1
(i ћ)( ∂_{t} + φ' ) ~ ic(i ћ)(M) ±[1 + (∇ +a' )^{2}/2 i^{2}(M)^{2}]
(i ћ)( ∂_{t} + φ' ) ~ ±[ic(i ћ)(M) + ic(i ћ)(M)(∇+a' )^{2}/2 i^{2}(M)^{2}]
(i ћ)( ∂_{t} + φ' ) ~ ±[c(i^{2} ћ)(M) + c( ћ)(∇+a' )^{2}/2(M) ]
remember M = m_{o}c/ћ
(i ћ)( ∂_{t} + φ' ) ~ ±[c(i^{2} ћ)(m_{o}c/ћ)+c( ћ)(∇ +a' )^{2}/2(m_{o}c/ћ)]
(i ћ)( ∂_{t} + φ' ) ~ ±[c(i^{2})(m_{o}c) +(ћ)^{2}(∇ +a' )^{2}/2(m_{o})]
(i ћ)( ∂_{t} + φ' ) ~ ±[(m_{o}c^{2}) +(ћ)^{2}(∇ +a' )^{2}/(2m_{o})]
remember A'_{EM} = iq/ћA_{EM}
(i ћ)( ∂_{t} + iq/ћφ ) ~ ±[(m_{o}c^{2}) +(ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}]
(i ћ)( ∂_{t} ) + (i ћ)(iq/ћ)(φ) ~ ±[(m_{o}c^{2})+ (ћ)^{2}(∇ + iq/ћa)^{2}/2m_{o} ]
(i ћ)( ∂_{t} ) + (i^{2})(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}]
(i ћ)( ∂_{t} ) ~ (qφ )±[(m_{o}c^{2})+ (ћ)^{2}(∇ + iq/ћa)^{2}/2m_{o} ]
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 nonrelativistic 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 nonrelativistic 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 lowenergy systems
Arbitrarily chose vector potential a = 0
Or keep it around for a nearPauli 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 4Vectors Reference for more on this.
Now, let's examine something interesting...
∂·∂ = (m_{o}c / ћ)^{2}: KleinGordon 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 the4Divergence of any 4Vector (even a Complexvalued one) must be Real
so∂·(P) = 0 : The conservation of 4Momentum (i.e. energy&momentum) for our KleinGordon relativistic particle.
This is also the equation of continuity which leads to the probability interpretation in the Newtonian limit.
R = (ct,r)  Location of an event (i.e. a particle)within spacetime 
U = dR/dτ  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 KleinGordon equation describes a nonlocal wave function, which "violates relativistic causality when used to describe particles localized to within more than a Compton wavelength,..."Baym. The nonlocality 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 waveparticle 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
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