# Gauge Invariance of superconducting circuits wavefunction model

Effective wavefunction describing the flow of superconducting elections (Cooper pairs):

Schrödinger equation for :

--- (3.5)

I will show that this model is invariant under the gauge transformation:

[Proof]

(3.5) is equivalent to

--- (1)

A simple calculation shows the following commutation relations.

Hence we have:

--- (2)

--- (3)

By using (2), we have:

--- (4)

Similarly, by using (3), we have:

Using (3) again:

--- (5)

From (4) and (5), (1) is equivalent to:

Now we can show that the phase defined by the following equation is gauge-invariant:

--- (3.16)

First, in terms of the phase , the gauge transformation of can be written as:

So, by combining with the gauge transformation of , the combination becomes gauge-invariant. By integrating it through a path from to , the following combination gives a gauge-invariant phase up to a constant .

By setting , we get the expression (3.16)