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Suppose φ is solution to Klein-Gordon equation, Multiplying it by -iφ* we get

[itex]iφ^*\frac{\partial^2φ}{\partial t^2}-iφ^*∇^2φ+iφ^*m^2=0[/itex] ..............(5)

Taking the complex conjugate of the Klein-Gordon equation and multiplying by -iφ we get

[itex]iφ\frac{\partial^2φ^*}{\partial t^2}-iφ∇^2φ^*+iφm^2=0[/itex]]..............(6)

If we subtract the second from the first we obtain

[itex]\frac{\partial}{\partial t}[i(φ^*\frac{\partial φ}{\partial t}-φ\frac{\partial φ^*}{\partial t})]+ ∇. [-i(φ^*∇φ-φ∇φ^*)]=0[/itex]...............(7)

This has the form of an equation of continuity

[itex]\frac{\partial p}{\partial t}+ ∇.j = 0[/itex].........................(8)

Source: https://www2.warwick.ac.uk/fac/sci/physics/staff/academic/boyd/stuff/dirac.pdf

Qns 1: What does it means to have the complex conjugate of the KL equation? I know we obtain eqn(5) by multiplying the KL eqn with -iφ*, so that we can get the probability but why do we need to come up with the complex conjugate of the KL eqn which lead us to eqn(6)?

Qns 2:Subsequently what is the rationale behind subtracting the second from the first to get eqn (7)?

Qns 3: Isn't eqn(8) conservation of charge? How does it relates to the KL eqn which is obtain from the energy-momentum conservation eqn?

[itex]iφ^*\frac{\partial^2φ}{\partial t^2}-iφ^*∇^2φ+iφ^*m^2=0[/itex] ..............(5)

Taking the complex conjugate of the Klein-Gordon equation and multiplying by -iφ we get

[itex]iφ\frac{\partial^2φ^*}{\partial t^2}-iφ∇^2φ^*+iφm^2=0[/itex]]..............(6)

If we subtract the second from the first we obtain

[itex]\frac{\partial}{\partial t}[i(φ^*\frac{\partial φ}{\partial t}-φ\frac{\partial φ^*}{\partial t})]+ ∇. [-i(φ^*∇φ-φ∇φ^*)]=0[/itex]...............(7)

This has the form of an equation of continuity

[itex]\frac{\partial p}{\partial t}+ ∇.j = 0[/itex].........................(8)

Source: https://www2.warwick.ac.uk/fac/sci/physics/staff/academic/boyd/stuff/dirac.pdf

Qns 1: What does it means to have the complex conjugate of the KL equation? I know we obtain eqn(5) by multiplying the KL eqn with -iφ*, so that we can get the probability but why do we need to come up with the complex conjugate of the KL eqn which lead us to eqn(6)?

Qns 2:Subsequently what is the rationale behind subtracting the second from the first to get eqn (7)?

Qns 3: Isn't eqn(8) conservation of charge? How does it relates to the KL eqn which is obtain from the energy-momentum conservation eqn?

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