Physics Guides

Introduction

The Gauge principle is one of the most powerful principles in modern theoretical physics. It can be used to usher in the four fundamental forces of nature and potentially even unify them.

The term Gauge was originally used by Herman Weyl in an attempt to unify General Relativity (GR) with Electromagnetism.¹ Weyl noticed that GR was invariant under changes of ‘gauge’ or length scale. However while GR can be described geometrically, Electromagnetism cannot, so his attempt at unification failed. Weyl continued to use the word gauge to describe physical invariance even though it no longer had anything to do with length scales.¹

Gauge and Electromagnetism

A simple example of gauge invariance can be understood by considering an electrical circuit (figure 1)

Figure 1. Changing the potential globally does not alter the circuit behaviour.

By changing the potential in this circuit by 10 volts everywhere, the overall behaviour of the circuit has not changed. The circuit is said to be invariant with respect to global transformations to the electric potential.

Gauge invariance is thereby used to describe physical symmetries such as the universe’s invariance with respect to position in space time. This means if you perform a given experiment here today and in the vicinity of Betelgeuse a year from now, you will expect to get the same results. There are many examples of gauge invariance; here we will take a close look at Electromagnetism.

The theory of Electromagnetism describes two physical fields, the electric field E and the magnetic field B. From these two fields we can define associated potentials, the scalar potential and vector potential A, given by:

	(1)
	(2)

Since the E and B fields are obtained via derivatives of both and A, addition of constants to these potentials leaves the fields unchanged. This is called Gauge Freedom; there are an infinite amount of potentials that can represent the same field. This property can be exploited to make a choice of gauge (a particular and A) that simplifies calculations. Changing the potential by a constant is called a Gauge Transformation of the first kind (a global transformation). It is also possible to make Gauge Transformations of the form:

	(3)
	(4)

Where f = f(x,t) is a continuous function. This is called a Gauge Transformation of the second kind. Since f is a function of x (space) and t (time), this is now a local transformation that can vary over space and time. The potential can be transformed arbitrarily, as long as the new ‘gauge’ satisfies equations (1) and (2) of the original E and B fields. This may lead us to question the physicality of the potential; if the choice of representation is arbitrary then perhaps the potential is just a convenient mathematical artifice that describes the ‘physical’ field. However, this view is incorrect, as can be observed via the Aharonov-Bohm effect.

Figure 2. Setup of the Aharonov-Bohm effect.¹

The Aharonov-Bohm effect is the phenomenon of an electron diffraction pattern been shifted by what appears to be a non-local field. Starting with the normal double slit diffraction setup, a solenoid is placed between the two slits (see figure 2). With no current in the solenoid, the usual diffraction pattern is observed. When there is current flowing in the solenoid the electron diffraction pattern is shifted horizontally. Having a current in the solenoid induces a magnetic field on the interior of the coil, directed perpendicular to the plane of the page, and zero outside the coil. This appears to be action at a distance, as there is no local field to provide a force that shifts the electron diffraction pattern. In fact the vector potential outside the solenoid may be non-vanishing and merely curl-free, which will still give B = 0 via equation (1). Dirac showed that an electron travelling in a potential will pick up an extra phase factor, which changes the positions of the maxima and minima in the diffraction pattern. This effect experimentally verifies that it is incorrect to consider a potential being any less physical than the field it is derived from.

As introduced above, gauge transformations of the second kind are known as local transformations as they can vary in space and time. This type of transformation in intimately connected with the four fundamental forces of nature. Here we will consider in detail the electromagnetic force. Consider a particle with wavefunction ψ, which evolves in time according to the Schrödinger equation:

(5)

Where all symbols are as usually defined. While this describes the physical evolution of the wavefunction over time, what we actually observe is the probability density |² changing. It can be trivially shown that the observable probability of the wavefunction is invariant under transformations of the form:

(6)

Where θ is a constant and thus this is a global transformation. This changes the wavefunction’s phase by an amount θ. Under this transformation the wavefunction will still satisfy the Schrödinger equation. Yang and Mills questioned the validity of considering global transformations, since the speed of light prohibits instantaneous global transformations. This sent them down another path, what if instead θ = θ(x) (i.e. a local transformation is made)? The wavefunction would simply no longer satisfy the Schrödinger equation. This is because the Laplacian derivative will now include an extra term due to the transformation’s dependence on x. To maintain gauge invariance we must modify the momentum operator in the Schrödinger equation by introducing the covariant derivative²:

(7)

Where e is the charge on the particle and A_µ =_µθ. Using equation (7), we can re-write the Schrödinger equation with natural units as:

Where contains the coefficients of each of the 4-vector derivatives. Proceeding on by making the transformation:

Gives:

Making the substitution for A_µ and expanding:

Cancelling the terms leaves:

With e=1 (natural units) the Schrödinger equation is recovered:

As shown above, by using the covariant derivative in place of the momentum operator, will now solve the Schrödinger equation when under local phase transformations. This was at the cost of introducing a non-zero, which is the electromagnetic 4-vector potential: