Massless particles and the Neutrino

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This page is under construction by Hartwig Poth, hupoth@gmail.com

For the four potential theory of gravitation, cf. Four potential theory of gravitation, a quantum particle, the 'gravon', can be found. It is massless and is defined by the following four vector wave function  Φ {\displaystyle \mathbf {\Phi } }

( 1 ) Φ = ( Φ 0 , Φ -> ) = ( Φ 0 , Φ x , Φ y , Φ z ) = ( − E , k x , k y , k z ) exp ⁡ ( i ( E t − k x x − k y y − k z z ) ) {\displaystyle (1)~~~\mathbf {\Phi } =(\Phi _{0},{\vec {\Phi }})=\left(\Phi _{0},\Phi _{x},\Phi _{y},\Phi _{z}\right)=\left(-E,k_{x},k_{y},k_{z}\right)\exp \left(i(Et-k_{x}x-k_{y}y-k_{z}z)\right)}

wherein  Φ {\displaystyle \mathbf {\Phi } } is the gravitational four potential [1]. The gravon has obviously the spin 0.

With (1) we obtain  ( 2 ) ∂ t Φ 0 = ∇ -> Φ -> {\displaystyle (2)~~~\partial _{t}\Phi _{0}={\vec {\nabla }}{\vec {\Phi }}}

This is a linear wave equation for the gravon and also a continuity equation; it is furthermore the third Maxwell type field equation for the four potential gravitation[2].

In analogy to (1) a photon  ϕ {\displaystyle \phi } can be defined by

( 3 ) ϕ = ( 1 , s x , s y , s z ) exp ⁡ ( i ( − E t + k x x + k y y + k z z ) ) {\displaystyle (3)~~~\phi =(1,s_{x},s_{y},s_{z})\exp \left(i(-Et+k_{x}x+k_{y}y+k_{z}z)\right)}

wherein  ( s x , s y , s z ) {\displaystyle (s_{x},s_{y},s_{z})} is the spin vector for the spin 1 of the photon; the spin vector behaves like an ordinary spatial vector and is directed along the momentum  ( k x , k y , k z ) {\displaystyle (k_{x},k_{y},k_{z})} of the photon. The time like component  s 0 {\displaystyle s_{0}} of the complete spinor  ( 1 , s x , s y , s z ) {\displaystyle (1,s_{x},s_{y},s_{z})} in (3) is  1 {\displaystyle 1} . The linear wave equation for this photon is  ( 4 ) ∂ t 1 exp ⁡ ( i ( − E t + k x x + k y y + k z z ) ) = − ( s x , s y , s z ) ∇ -> exp ⁡ ( i ( − E t + k x x + k y y + k z z ) ) {\displaystyle (4)~~~\partial _{t}1\exp \left(i(-Et+k_{x}x+k_{y}y+k_{z}z)\right)=-(s_{x},s_{y},s_{z}){\vec {\nabla }}\exp \left(i(-Et+k_{x}x+k_{y}y+k_{z}z)\right)}

That equation (4) allows for example to apply the four potential of gravitation  ( Φ 0 , Φ -> ) {\displaystyle (\Phi ^{0},{\vec {\Phi }})} in analogy to the influence of the common electromagnetic four potential onto the Dirac electron[3]

( 5 ) i ℏ ∂ ψ ∂ t = [ c α -> ( p -> − m 0 Φ -> ) + m 0 c 2 Φ 0 + c 2 β m 0 ] ψ {\displaystyle (5)~~~i\,\hbar {\frac {\partial \psi }{\partial t}}=\left[c\,{\vec {\alpha }}\,\left({\vec {p}}-m_{0}\,{\vec {\Phi }}\,\right)+m_{0}\,c^{2}\,\Phi ^{0}+\,c^{2}\beta \,m_{0}\right]\psi }

to the propagation of the photon

( 6 ) ∂ exp ⁡ ( i ( − E p h o t o n t + k x x + k y y + k z z ) ) ∂ t s 0 = [ ( p -> − E p h o t o n c 2 Φ -> ) ( s x , s y , s z ) + E p h o t o n Φ 0 ] exp ⁡ ( i ( − E p h o t o n t + k x x + k y y + k z z ) ) {\displaystyle (6)~~~{\frac {\partial \exp \left(i(-E_{photon}t+k_{x}x+k_{y}y+k_{z}z)\right)}{\partial t}}s_{0}=\left[\left({\vec {p}}-{\frac {E_{photon}}{c^{2}}}\,{\vec {\Phi }}\,\right)(s_{x},s_{y},s_{z})+E_{photon}\,\Phi ^{0}\right]\exp \left(i(-E_{photon}t+k_{x}x+k_{y}y+k_{z}z)\right)}

When we suppose that  Φ -> {\displaystyle {\vec {\Phi }}} vanishes, if the source of gravitation is virtually at rest and if we consider for simplicity a photon which propagates along the  z {\displaystyle z} -axis with the momentum  k z {\displaystyle k_{z}} we obtain eventually

( 7 ) E p h o t o n = k z ( 1 + Φ 0 ) {\displaystyle (7)~~~E_{photon}=k_{z}(1+\Phi ^{0})}

and thus for the respective energies of the photon at positions  z 1 {\displaystyle z_{1}} and  z 2 {\displaystyle z_{2}} an energy difference  Δ E {\displaystyle \Delta E}

( 8 ) Δ E = Φ 0 ( z 1 ) − Φ 0 ( z 2 ) {\displaystyle (8)~~~\Delta E=\Phi ^{0}(z_{1})-\Phi ^{0}(z_{2})}

This result is already known from the general theory of relativity [4].

1. ↑ 'Massless particles and the Neutrino', from Hartwig Poth, published on 11.01.2022 under ISBN 9783844243321
2. ↑ 'Four Potential Gravitation and its Quantization', from Hartwig Poth, published on 05.04.2014 under ISBN 978-3-8442-9165-0
3. ↑ 'Four Potential Gravitation and its Quantization', chapter 13 on page 29 equation (191), from Hartwig Poth, published on 05.04.2014 under ISBN 978-3-8442-9165-0
4. ↑ Torsten Fließbach, Allgemeine Relativitätstheorie (Spektrum Akademischer Verlag, Heidelberg Berlin, 4. Auflage ISBN3-8274-1356-7), 12 Photonen im Gravitationsfeld p.62