The term “Configuration” in quantum field theory, what does it mean?
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I would like a thorough explanation of what the word “configuration”, means, as used in ‘Quantum Field Theory’.
I have seen the word used in various phrases, such as “the field configuration φ(x)”, “given initial configuration” and possibly “integrate over all configurations”.
I have some knowledge of non-relativistic quantum theory.
Could your answer include the use of the idea “configuration” being used in various equations starting with something simple? Perhaps you could refer me to where I might find such an answer?
I have seen the word used in various phrases, such as “the field configuration φ(x)”, “given initial configuration” and possibly “integrate over all configurations”.
I have some knowledge of non-relativistic quantum theory.
Could your answer include the use of the idea “configuration” being used in various equations starting with something simple? Perhaps you could refer me to where I might find such an answer?
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A field configuration is simply a particular set of values for the field at every point in space. For example, a particular configuration for the 0+1-dimensional field might be
, or , or even some arbitrarily complex form that isn't describable with elementary functions.
The point is that a field configuration is simply a "point in function-space". The exact definition of "function-space" imposes restrictions on the kind of configurations such a field can adopt (for example, due to UV cutoffs, QFTs cannot admit large fluctuations on very small length scales; we also typically ask that the configurations vanish at infinity or conform to some other boundary condition). When you integrate over all field configurations, you are integrating in function-space over all allowed points, or equivalently integrating over all of the functions that your field could possibly be.
, or , or even some arbitrarily complex form that isn't describable with elementary functions.
The point is that a field configuration is simply a "point in function-space". The exact definition of "function-space" imposes restrictions on the kind of configurations such a field can adopt (for example, due to UV cutoffs, QFTs cannot admit large fluctuations on very small length scales; we also typically ask that the configurations vanish at infinity or conform to some other boundary condition). When you integrate over all field configurations, you are integrating in function-space over all allowed points, or equivalently integrating over all of the functions that your field could possibly be.
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