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- How are phi and psi (solutions to the Klein-Gordon and Dirac equations) expressed mathematically in quantum field theory?

I found a copy of David McMahon's "Quantum Field Theory Demystified" and I'm already confused on page 4 where he says, " . . in order to be truly compatible with special relativity, we need to discard the notion that [itex]\phi[/itex] and [itex]\psi[/itex] in the Klein-Gordon and Dirac equations respectively describe single particle states. In their place, we propose the following new ideas:

— The wave functions [itex]\phi[/itex] and [itex]\psi[/itex] are not wave functions at all, instead they are fields.

— The fields are operators that can create new particles and destroy particles."

As i understand things,

— the [itex]\psi[/itex] in the Schrodinger equation represents a complex number at every point in space and time, while in the Dirac equation represents four complex numbers at every point in space and time. (I don't know what the [itex]\phi[/itex] in the Klein-Gordon equation represents, but I'm guessing something similar.)

— an operator is something that changes a function into a different function. One way to think about it is - if a function is a vertical list of n complex numbers, then an operator is an nxn matrix that can be multiplied by the column of n numbers to produce a different column of n numbers.

In quantum field theory, what exists at every point in space and time? A matrix? More than one matrix?

— The wave functions [itex]\phi[/itex] and [itex]\psi[/itex] are not wave functions at all, instead they are fields.

— The fields are operators that can create new particles and destroy particles."

As i understand things,

— the [itex]\psi[/itex] in the Schrodinger equation represents a complex number at every point in space and time, while in the Dirac equation represents four complex numbers at every point in space and time. (I don't know what the [itex]\phi[/itex] in the Klein-Gordon equation represents, but I'm guessing something similar.)

— an operator is something that changes a function into a different function. One way to think about it is - if a function is a vertical list of n complex numbers, then an operator is an nxn matrix that can be multiplied by the column of n numbers to produce a different column of n numbers.

In quantum field theory, what exists at every point in space and time? A matrix? More than one matrix?

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