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The prospect exists that quantum mechanics and relativity are really two aspects of the same thing. Both of these domains of physical observation rely upon a constant which defines a projective bundle. In a categorical sense we may find that quantum physics and relativity are equivalent, and further they embed within a larger system.

Relativity has a projective subspace defined by null...

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The prospect exists that quantum mechanics and relativity are really two aspects of the same thing. Both of these domains of physical observation rely upon a constant which defines a projective bundle. In a categorical sense we may find that quantum physics and relativity are equivalent, and further they embed within a larger system.

Relativity has a projective subspace defined by null rays. A projective space is constructed from a space by looking at all lines through a point and considering them as equivalent element under all rescalings. Lorentz spacetime contains this in a set of light rays which pass through a point. The distance along a null ray is zero, no matter how much one rescales coordinates x or t. This then constructs the projective Lorentz group, where the Lorentz group has a projective fibre or line bundle over this projective variety.

Physically the speed of light is just a conversion factor which converts between distance and time. The constancy of light speed means this factor defines the projective subspace in spacetime.

Quantum mechanics has a similar structure. The Planck constant ħ is a conversion factor between the variance in position and momentum with the Heisenberg uncertainty principle ΔpΔx ~ ħ. In addition there is a projective variety. To show this we can start with the Schrodinger equation:

iħ|∂ψ)/∂t = H|ψ)

I have to use parentheses for bra-ket notation on this blog. We look at the overlap between the state |ψ(t)) and |ψ(t+δt)) and Taylor expand it

(ψ(t)|ψ(t+δt)) = (ψ(t)|ψ(t)) + δt(ψ(t)|∂ψ(t)/∂t) + (δt^2/2)(ψ(t)|∂^2ψ(t)/∂t^2) + …

Now use the Schrodinger equation to get

(ψ(t)|ψ(t+δt)) = (ψ(t)|ψ(t)) – (i/ħ)δt(ψ(t)|H|ψ(t)) – (1/ħ^2)(δt^2/2)(ψ(t)|H^2|ψ(t)) + …

We now evaluate the modulus square of this to get

|(ψ(t)|ψ(t+δt))^2 = 1 + (δt/ħ)^2 [(ψ(t)|H^2|ψ(t)) - (ψ(t)|H|ψ(t))^2] + …

This term O(δt^2) can be written as (δt/ħ)^2 ΔH^2, which recovers the Heisenberg uncertainty principle. The overlap is determined by a phase term e^[(δt/ħ)^2 ΔH^2], which is the line bundle or projective bundle.

The only known intertwining relationship between ħ and c is the Planck length, or other Planck scales

L_p = sqrt{Għ/c^3}

This may be written as ħ/c = (L_pc)^2/G, which can be expressed according to the Planck mass m_p = sqrt{ħc/G} as

ħc = Gm_p^2 = 3.16x10^{-26)kg m^3/s^2.

This is then a constant, which is in naturalized units “one.”

Cheers LC

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