Wednesday, 27 June 2012

Scalar Fields

Beginning this blog on the "s" theme of its title, I want to talk about scalar fields, or scalars. A scalar field is something that has a value at every point in space: the canonical example is temperature. There is a value, but no direction.

A magnetic field on the other hand, points from north to south, it has a direction and is known as a vector field. If I look at the Earth the "normal" way up, I see this as going down on the map, but I can look the Earth the other way up, because there is nothing special about the choice of map makers, and I see the magnetic field going up. Vectors care about how you look at them. They care about the frame of reference. Scalars don't care what direction you look at them or how fast you are moving: they are Lorentz invariant.

For this reason, scalars are odd things, as they don't transform under the Lorentz symmetry of special relativity. This allows them to have a vacuum expectation value, or vev. A vev means that the scalar field has a value that is the same everywhere in space. This is not something a vector field could have without picking out a preferred direction in space, and violating a form of the cosmological principle. The vev has an energy, and it contributes to the cosmological constant, which effects the expansion rate of our universe.

Interestingly, this is contrary to the point of this video explaining the Higgs (which I found on Cosmic Variance). The video explains quantum fields as things whose ripples are what we observe: they are everywhere but we only see them when they ripple. However, the Higgs is a scalar, and its vev also contributes to the cosmological constant, which changes when the electroweak symmetry is broken in the early universe. With scalars, we see more than just their ripples.

I mentioned above that temperature is the canonical example to explain what a scalar field is to the non-specialist. However, in relativity, we express the temperature as the energy density of the electromagnetic field, but the energy density changes when we move to different frames of reference. Temperature is the fourth root of the energy density, which forms part of a four vector and transforms under Lorentz transformations. Temperature is NOT a scalar in relativity! In Cosmology a major number is the temperature of the CMB. How can we speak of this in a way that does not violate the cosmological principle? The truth is that we only measure the temperature of the CMB in the preferred frame of reference that is at rest with respect to the CMB. The CMB defines the preferred frame. Models where we mess with this are called "tilted universes" (see e.g. Liddle and Lyth's book on Inflation).

When we write down fundamental theories we don't want them to care about such arbitrary choices like the maps of the Earth. Scalars fit this bill. If we want our fundamental theories to contain vector fields or other more complicated objects, however, we must "contract" them up to make scalars. Being a scalar is very important and is related to fundamental concepts in physics about symmetries and the action principle.

In group theory, scalars are the singlets of any group. A representation every group shares. They normally don't feel whatever force is associated with a group, for example neutrinos are singlets under the U(1) of electromagnetism (EM), so they are not charged. but neutrinos are not scalar fields, we name things "scalars" in the sense of "scalar field" by the way the field behaves under the symmetries of General Relativity (GR), the diffeomorphisms. However, because GR sees all energy density, even scalars of diffeomorphisms source gravity, in a way that scalars of EM do not source EM.

Fundamental scalars are very odd things. By which I mean things that are scalars without us having to "make" them that way, in the way we build scalars in EM to make the Maxwell Lagrangian by contracting the vector fields. The only one that we think exists in the standard model is the Higgs field, and an aversion to fundamental scalars is one philosophical motivation for theories like technicolor that are alternatives to the Higgs. The scalars of technicolor are not fundamental, they are "composite" scalars. Like the pion of the strong force, they are built by adding two fields of opposite spin.

However,  one theory that is full of scalar fields is string theory. These scalars are describing things to do with the internal space that we lowly 4d mortals can't see (see for example These fields are called moduli. These apparently fundamental scalars in 4d are just vestiges of higher dimensional gravity, and appear to us after the famous phenomenon of Kaluza-Klein compactification. String theory scalars are very important in making string theory reproduce the physics we know, give us lots to play with in the physics we don't, but are infamous as the source of the string landscape and the charge of loss of productivity often brought against the theory. (string theory also furnishes us with many other similar fields called axions, which are "pseudo-scalars" and behave differently under spatial reflections, but they are the subject of a whole other post…)

Another theory that contains lots of scalars is supersymmetry (SUSY). SUSY builds in scalars in what are called chiral multiplets. The chirality (or handedness) of the weak nuclear force is one of the most important facts about our universe and is very constraining for model builders. Incorporating SUSY into a theory that is chiral gives us all the superpartners as scalars or "sparticles" -  and at least as many as there are fermions (quarks and leptons) in the standard model. We know the weak force is chiral: if the world is also supersymmetric, then scalars play a key role.

But why, apart from the possible imminent detection of our first genuine scalar in CERN next week, am I telling you about scalars? I'm a cosmologist, and scalars are everywhere in cosmology theory. Why? Because they are easy to work with. One can often do away with the awkward phase space descriptions one needs to properly describe for example photons and neutrinos (the analogous distributions to the Maxwell-Boltzmann distribution in ordinary statistical mechanics). All modified gravity theories are GR plus a scalar or more. (the exception are the odd (and oddly named) "Galileons", also the subject of future post)

What do we use scalars for in cosmology? Well, scalars can have potentials. Because they are Lorentz invariant, any function of them is, and this is their great utility. They serve us as inflatons, dark energy, and dark matter. However, our greatest theoretical problem, the cosmological constant problem, is intimately related to these very potentials, and the vevs that allow scalar fields to perform the Higgs mechanism. (if you've never read it, I highly recommend Weinberg's 1989 review)

Do true scalars exist? Because of the need for a change in, or constant, vacuum energy in any theories of electroweak symmetry breaking (what the Higgs does), inflation, or Dark Energy/modified gravity, one always requires at least an approximate, or composite, scalar degree of freedom: technibaryons, or massive gravitons are all described by effective scalar degrees of freedom. One could even argue that string theory, in its true 10 or 11 dimensional form doesn't have scalars: the effective ones are degrees of freedom of a string. The question of whether we have fundamental scalars is still open, but could come one step closer to being resolved on Wednesday at CERN. Even if particle physicists find the Higgs, knowing whether it is really fundamental will be a whole other game. Although the LHC should be able to distinguish it from many popular technicolor theories, we will probably have to wait longer to find out. (I'm sure Jester will have plenty to say about this)

Finally, even in a strongly coupled theory like technicolor, compositeness is not a definite concept thanks to the hot topic of holography and dualities. In dual theories the "fundamental" fields swap roles, so composites on one side of the duality are fundamental from the other point of view.

1 comment:

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