Objects Involved in a Transformation

When quantum field theory textbooks discuss transformations, they are usually describing the relation between fields and coordinates. Tong’s lecture notes, for example, say:

we are dealing with an active transformation in which the field is truly shifted.

If we were instead dealing with a passive transformation in which we relabel our choice of coordinates.

Descriptions of this kind conceal an object that is rarely named explicitly: spacetime.

If fields and coordinates were the only objects involved, then shift field and relabel coordinates would not differ in substance; they would merely be two views of the same relative motion.

Once spacetime is included explicitly, we can discuss active and passive transformations more carefully, as well as why spacetime is so often omitted from the discussion.

Active Transformations

An active transformation changes the distribution of a field over spacetime. For a real scalar field 𝜙, let spacetime be a manifold 𝑀. An active transformation may then be represented as

Here 𝜙′(𝑝′) =𝜙(𝑝): the value of the transformed field 𝜙′ at 𝑝′ equals the value of the original field 𝜙 at 𝑝.

It is important that neither the spacetime manifold nor its coordinate system changes under an active transformation. What changes is the field configuration.

For example, a Lorentz transformation sends 𝜙 ↦𝜙′, where 𝜙′(𝑝′) =𝜙(𝑝) and

A common source of confusion is that this expression looks exactly like a Lorentz coordinate transformation. In the active picture, however, it is not a coordinate transformation. It relates the coordinates of 𝑝′ to those of 𝑝, with the field taking the same value at these two points before and after the transformation.

In other words, the object acted on by the “Lorentz transformation” here is the field configuration, not the coordinate system.

Passive Transformations

A passive transformation changes the coordinate system on spacetime. It changes the coordinates assigned to the same spacetime point from 𝑥𝜇 to 𝑥′𝜇:

For a Lorentz transformation,

Neither spacetime nor the field configuration changes under a passive transformation. This is the transformation commonly meant when we speak of a “Lorentz transformation.”

Comparing Active and Passive Transformations

To compare the two descriptions more carefully, introduce the coordinate map 𝑥:

Then 𝑥𝜇 =𝑟𝜇 ∘𝑥. Introduce ˜𝜙 such that

or equivalently 𝜙 =˜𝜙 ∘𝑥.

The active transformation can therefore be written as

Now introduce

The component expression

may then be written as

Define ˜𝜙′ ∘𝑥 =𝜙′. It follows that

Suppressing 𝑝 and the tilde, and writing composition with the map simply as 𝛬𝑥, gives the familiar expression

If 𝛬−1 denotes the inverse of 𝛬—without proving its existence here—then

Therefore,

Again suppressing 𝑝 and the tilde, and writing composition as 𝛬−1𝑥, yields

For a passive transformation, spacetime and the field configuration remain unchanged:

The coordinate system changes, meaning that the map 𝑥 is replaced by 𝑥′ such that

Define

Then

Suppressing 𝑝 and the composition symbol gives

From 𝑥′(𝑝) =𝛬 ∘𝑥(𝑝), we obtain

Therefore,

Suppressing 𝑝, the composition symbol, and the prime on 𝑥 gives

Why Is Spacetime Often Omitted?

The background spacetime of quantum field theory is Minkowski spacetime. Its high degree of symmetry makes spacetime itself easy to overlook: applying a transformation to the field configuration and then making the corresponding change of coordinates produces a result equivalent to the original description.

In fact, both the field configuration and the coordinates have changed relative to spacetime. The symmetry of Minkowski spacetime prevents that change from becoming visible. On an arbitrary spacetime, this equivalence no longer holds.

Equivalence of Active and Passive Transformations in Minkowski Spacetime

In Minkowski spacetime, appropriate active and passive transformations can be chosen to have the same final effect: the same field value is associated with the same spacetime coordinate, or equivalently, the coordinate-dependent functions used to describe the field configuration are identical.

Again consider a Lorentz transformation. The active transformation gives

The passive transformation gives

Therefore,

Moreover,

Thus the field-configuration function ˜𝜙′ obtained from the active transformation and the function ¯𝜙′ obtained from the passive transformation take the same value at the same argument. The field-configuration function alone therefore does not reveal whether it arose from an active or a passive transformation.

Active and Passive Transformations Are Generally Inequivalent on an Arbitrary Spacetime

On an arbitrary spacetime, an active transformation changes the field configuration assigned to spacetime points. Different spacetime points may have different curvature, and curvature distinguishes one point from another. An active transformation therefore changes the correspondence between curvature and field value.

A passive transformation changes neither spacetime nor the field configuration on it, so it does not change the correspondence between curvature and field value.

From this perspective, active and passive transformations cannot in general be equivalent.