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(2.4.1) and (2.8.2) to describe to same quantum system. Thus the Shrodinger equation is a many-to-one description of quantum dynamics.

Similarly, the Lagrangian (2.4.3) is also a many-to-one description of quantum systems. Underthe transformation (2.8.1), L in eqn (2.4.3) is changed into

L(d

dt|, |) = (t)| i

d

dt (H )|(t) (2.8.3)

L and L describe the same quantum system. We note that the difference of the two LagrangiansL L = | = is a total time derivative.

2.8.3 **The gauge invariance

The result in the section 2.3.3 can be easily generalized to three dimensions. A charge-1 particlemoving in an electric field E and a magnetic field B is described by the following equation ofmotion

mxi = Ei + ijk xjBk (2.8.4)

If we introduce potential (x, t) and vector potential A(x, t) to express E and B:

Bi = ijkjAk, Ei = Ai i,

then the equation of motion (2.8.4) can be derived from the following coordinate-space Lagrangian

L =1

2m(xi)2 + Aixi. (2.8.5)

The corresponding phase-space Lagrangian is given by

L = pixi

1

2m(pi A

i)2 (2.8.6)

However, the representation of the electromagnetic field (Ei, Bi) in terms of (, Ai) is notunique. (, Ai) and (, Ai) give rise to the same electromagnetic field if they are related by

Ai(x, t) = Ai(x, t) i(x, t), (x, t) = (x, t) + t(x, t). (2.8.7)

So (, Ai) is a many-to-one representation of the physical electromagnetic field (Ei, Bi). Thetransformation (2.8.7) between equivalent representations is called gauge transformation.

We have seen that the Lagrangian is a many-to-one representation of classical systems. The state vector|(t) is many-to-one representation of quantum state and quantum motion. We can call a transformationbetween any knids of equivalent representations a generalized gauge transformation. So L L + d/dt and|(t) e(t)|(t) are two examples of generalized gauge transformations.

We can write the equation of motion in terms of (, Ai):

mxi = (Ai i) + (iAj jA

i)xj (2.8.8)

Clearly, such an equation of motion is variant under the gauge transformation, indicating that theequivalent (, Ai)s give rise to the same motion.

Under the gauge transformation (, Ai) (, Ai), the coordinate-space Lagrangian (2.8.5) ischanged to

L L =m

2(xi)2 + Ai xi = L t ix

i = L d

dt

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We know that two Lagrangians differ by a total time derivative give rise to the same equation ofmotion and describe the same classical system. So in this sense, we may say that the coordinate-space Lagrangian is gauge invariant (up to a total time derivative).

However, the phase-space Lagrangian (2.8.6) is not invariant under the gauge transformation!The change is not even a total time derivative. As a result, the phase-space equation of motion

pi = i +pj A

j

m iAj

, xi

=pi A

i

m

is not gauge invariant. However, this discussion is misleading. In fact the phase-space Lagrangian(2.8.6) is also gauge invariant, provided that we allow the momentum pi to change under the gaugetransformation. If pi transform as

pi pi = pi + i (2.8.9)

then the phase-space Lagrangian is gauge invariant:

L(pi, xi; Ai, ) = L(pi, x

i; Ai, )

(up to a total time derivative.)

This result is very interesting. For a particle in magnetic field, the canonical momentum is notgauge invariant and is unphysical. Thus is it inconvenient (or even incorrect) to use the coordinate-momentum pair (xi, pi) to describe a classical state. We cannot use (x

i(t), pi(t)) alone to specify aclassical motion. We must also specify the background (, Ai). Thus a classical motion is describedor labeled by (xi, pi),Ai . Two sets of notations (x

i, pi),Ai and (xi, pi),Ai describes the same

motion if (pi, , Ai) and (pi, , A

i) are related by eqn (2.8.9) and eqn (2.8.7).

To find a more convenient notation, let us introduce a gauge invariant quantity

Pi pi Ai

which will be called gauge invariant momentum. We can use the gauge invariant pair (xi, Pi) to

describe a classical state. The phase-space Lagrangian can be rewritten in terms of (xi, Pi)

L = Pixi

1

2m(Pi)

2 + Aixi (2.8.10)

Since + Aixi changes by a total time derivative under a gauge transformation, the phase-spaceLagrangian is gauge invariant up to a total time derivative. The equation of motion for (xi, Pi) canbe obtained from the stationary paths of above Lagrangian (see eqn (2.3.6)):

Pi = i + (iAj jA

i)xj , xi =Pim

(2.8.11)

The equation of motion is explicitly gauge invariant.We like to stress that, despite its name, Pi are not canonical momentum ofxi. It is pi, not Pi, that correspond

to i xi

in the quantum theory.

It is quite interesting to see that the ambiguity of the Lagrangian description of classical systems,the ambiguity of the state vector description of quantum system, and the gauge invariance inelectromagnetism have a such close relation. The phase ambiguity of the quantum state and thegauge invariance in electromagnetism are parts of foundations in our understanding of quantumphysics and electromagnetism. Their close connection may indecate a close connection betweenquantum physics and electromagnetism. The emergence of light discussed in chapter 6 may be amanifestation of such a connection.

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Problem 2.8.2

Consider a mass m = 1 and charge q = 1 particle in a uniform magnetic field in the z direction. The strength ofthe magnetic field is B = 1.(a) Assume at t = 0, the particle is in a state (p1, p2, p3; x1, x2, x3) = (1, 0, 0; 1, 1, 0). Find the pi(t) and x

i(t)for t > 0. Plot trajectory (x1(t), x2(t)) in the (x1, x2) plane and trajectory (x1(t), p1(t)) in the (x

1, p1) plane.(b) Assume at t = 0, the particle is in a state (P1, P2, P3; x

1, x2, x3) = (1, 0, 0; 1, 1, 0). Find the Pi(t) and xi(t)

for t > 0. Plot trajectory (x1(t), x2(t)) in the (x1, x2) plane and trajectory (x1(t), P1(t)) in the (x1, P1) plane.

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