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I have a question: Consider the path integral:

Z = DA exp[-iS(A)] = C exp [iW(J)]

where S(A) = $d^4x L.

Above, the field variable is A, the source of the field is J, the
amplitude is W(J), the action is S(A), the Lagrangian density is L, and
C is an overall product factor independent of J and often containing a
product of inverse determinants.

Let's posit that L=0, everywhere. We take that as a supposition.
Perhaps 0 = some other expression involving the sources and fields, but
nonetheless, this expression L is always = 0.

Would the following deductions be true / permissible?

1) S(A) = 0, because the integral over a volume of anything which is
zero, is itself zero. No constants of integration come into play. For
example, thinking about Maxwell's equations in integral form, if there
is a three volume within which the charge is zero everywhere, then the
total enclosed charge is zero.

if 1) is true, then:

2) Z = DA, because exp[-iS(A)] = 1

if 2) is true, then:

3) given Z=DA, we can select C such that DA = C. Then, exp [iW(J)] =1.

if 3) is true, then

4) W(J) = 2pi n and so is quantized.

Please evaluate and advise if there is any flaw in this logic.

Thanks,

Jay.
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Jay R. Yablon
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