Global vs. Local Symmetries: The Principle of Gauge Invariance

Global vs. Local Symmetry: An Intuitive Picture

Let's begin with a concept you are familiar with from quantum mechanics: the wave function of a particle, which is a complex-valued field. The absolute phase of this wave function is not observable; only relative phases are. This suggests a symmetry: we can change the phase of the wave function everywhere by the same amount, and the physical predictions will remain unchanged. This is a global phase symmetry.

Consider the classic double-slit experiment with electrons. The interference pattern observed on the screen depends on the relative phase of the electron waves arriving from each slit.

If we apply a uniform phase shift to the entire electron field, the relative phase between the two paths remains the same, and the interference pattern is unaffected. This is a global symmetry. The transformation parameter (the phase angle) is a constant throughout space and time.

Now, imagine we could apply a different phase shift to the electron wave depending on its position. For instance, what if the phase of the wave passing through the top slit was shifted differently from the wave passing through the bottom slit? This would change their relative phase upon recombination, altering the interference pattern. This is a local symmetry transformation, where the phase angle is a function of the spacetime coordinate, $\alpha (x)$. A simple theory of a free electron is clearly not invariant under such arbitrary local phase shifts.

Let's formalize this distinction.

As the text highlights, a global symmetry involves a transformation that is identical at all spacetime points. In contrast, a local (or gauge) symmetry involves a transformation whose parameters can vary independently at each spacetime point.

The Breakdown of Local Invariance

We have established that a typical theory of free particles, while possessing global symmetries, is not invariant under the corresponding local transformations. Let's see precisely why this happens.

Recall the Lagrangian for a free Dirac field, which describes a free electron:

This Lagrangian is invariant under a global U(1) phase transformation:

where $\alpha$ is a constant. The mass term, $\bar{\psi }\psi$, transforms into $\bar{\psi }{e}^{-iq\alpha }{e}^{iq\alpha }\psi =\bar{\psi }\psi$, which is trivially invariant. The kinetic term is also invariant because the constant phase factor $e^{iq\alpha }$ commutes with the derivative operator ${\partial }_{\mu }$.

Now, let's "gauge" this symmetry by promoting the constant parameter $\alpha$ to a function of spacetime, $\alpha (x)$. The transformation becomes:

The mass term is still invariant. However, the derivative in the kinetic term now acts on the product of $e^{iq\alpha (x)}$ and $\psi (x)$. Using the product rule:

When we substitute this into the kinetic term $i{\bar{\psi }}^{\prime }{\gamma }^{\mu }{\partial }_{\mu }{\psi }^{\prime }$, the desired transformation factor $e^{-iq\alpha (x)}$ from ${\bar{\psi }}^{\prime }$ will cancel the first part, but an unwanted extra term remains:

The symmetry is broken. The derivative ${\partial }_{\mu }$, which compares the value of the field at infinitesimally separated points $x$ and $x+dx$, is the source of the problem. Under a local transformation, the phase reference frame changes from point to point, so a simple derivative is no longer a meaningful comparison.

This problem and the path towards its solution are the main topics of the following video.

The Principle of Local Gauge Invariance

The failure of our free-particle Lagrangian to be locally symmetric presents a choice. We could conclude that nature simply doesn't possess such local symmetries. Or, we could take a more audacious path and elevate local symmetry to a fundamental principle. This is the Principle of Local Gauge Invariance: we postulate that the fundamental laws of physics must be invariant under local gauge transformations.

This is a profoundly powerful idea. If we insist on local symmetry, we are forced to modify our theory to restore it. To cancel the unwanted term $-q\bar{\psi }{\gamma }^{\mu }\psi ({\partial }_{\mu }\alpha (x))$, we must introduce a new component to our theory that transforms in just the right way. As we'll see in the next lesson, this requires two modifications:

1. Replacing the ordinary derivative ${\partial }_{\mu }$ with a covariant derivative $D_{\mu }$.
2. Introducing a new vector field, called a gauge field, which couples to the matter field. This coupling is the interaction.

The requirement of local gauge invariance, therefore, acts as a guiding principle that dictates the existence and form of interactions (forces) in our universe. By demanding that the U(1) phase symmetry of the Dirac Lagrangian be local, we are forced to introduce the electromagnetic field. The theory of quantum electrodynamics (QED) is, at its heart, the theory of a Dirac field with a local U(1) gauge symmetry.

The following text provides a compact and insightful summary of this motivation.

As Shapiro's notes suggest, the gauge field acts as a "connection," allowing us to compare the phase of the matter field at different spacetime points in a consistent way. The demand for local symmetry is not just a mathematical game; it's a profound physical principle that generates the structure of interactions from a symmetry requirement.

Conclusion

In this lesson, we have moved beyond the global symmetries that lead to conservation laws and into the more restrictive and creative realm of local symmetries.

Key Takeaways:

• A global symmetry is a transformation whose parameters are constant throughout spacetime (e.g., a constant phase shift $\psi \to e^{i\alpha }\psi$).
• A local (gauge) symmetry is one whose parameters are functions of spacetime coordinates (e.g., a position-dependent phase shift $\psi \to e^{i\alpha (x)}\psi$).
• Lagrangians for free fields, such as the Dirac Lagrangian, are typically invariant under global symmetries but not under the corresponding local symmetries. The failure point is the derivative term, which compares fields at different points.
• The Principle of Local Gauge Invariance is the postulate that fundamental theories must be invariant under local gauge transformations.
• This principle is not descriptive but prescriptive: it forces the introduction of new gauge fields that mediate interactions and compels the replacement of ordinary derivatives with covariant derivatives.

We have laid the conceptual groundwork. We've seen why local symmetry is a problem for simple theories and have motivated the principle that turns this problem into a powerful tool for theory construction. In our next lesson, we will put this principle into action. We will explicitly construct the U(1) gauge field and the covariant derivative needed to make the Dirac Lagrangian locally symmetric, and in doing so, we will derive the famous interaction term of Quantum Electrodynamics from first principles.