This board is full of shitty threads so I am going to try to

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This board is full of shitty threads so I am going to try to make a not shitty one.

ITT: We teach /sci/ something science or math related.

I am going to go through a basic explanation of what a Quantum Field is because I am pretty sure a decent bit of /sci/ is familiar with the basics of QM but not so much QFT.
Ok so a Quantum Field is essentially a sum of operators and wavefunctions.


Say we have a theory where the state vector [math] \left| {n\left( p \right)} \right\rangle [/math] represents the nth energy level of theory with momenta p.

Then we can define an annihilation operator, [math] {a_p} [/math], of the theory as an operator which lowers the energy of the state on which it acts.

i.e. [math] {a_p}\left| {n\left( p \right)} \right\rangle = c\left| {n\left( p \right) - 1} \right\rangle [/math] for some constant c.


A creation operator can be defined as the hermitian conjugate of the annihilation operator.

i.e. [math] {a^\dag }_p [/math] such that [math] {a^\dag }_p\left| {n\left( p \right)} \right\rangle = {c_2}\left| {n\left( p \right) + 1} \right\rangle [/math] for some constant c_2.


Creation and annihilation operators, of a bosonic theory, obey the following commutation relation. [math] \left[ {{a^\dag }_{{p_1}},{a_{{p_2}}}} \right] = \delta _{{p_2}}^{{p_1} [/math]

So for a simple wave function [math] {e^{ipx}} [/math] we can roughly define a quantum field: [math] \Phi \left( x \right) = \sum\limits_p {\left( {{a^\dag }_p{e^{ipx}} + {a_p}{e^{ - ipx}}} \right)} [/math]


Quantum Fields have the important property that they are an operator, or an observable.


A more accurate definition that fits any real scalar field, would be in the form of a momentum Fourier transform. i.e. [math] \Phi \left( x \right) = \int {\frac{{{d^3}p}}{{{{\left( {2\pi } \right)}^3}}}} \frac{1}{{\sqrt {2{E_p}} }}\left( {{a^\dag }_p{e^{ipx}} + {a_p}{e^{ - ipx}}} \right) [/math] where E_p is the energy of the field at momentum p.