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I understand the basic idea of these two concepts, but I have question about what they really "are" (a little hard for me to put into words). When one talks about the cosmological constant, it seems as though it is some intrinsic property of space like an innate energy density. But quintessence is described by a scalar field, unlike the cosmological constant, and it appears to be different in some way.

I think what I am really trying to wrap my head around is this idea of a scalar field. What is a scalar field really? Is it nothing more than a cute piece of mathematics to describe something that we don't understand at a more fundamental level? Or is there more to it? Forgive me if this doesn't make sense, but the idea of a scalar field is tossed around a lot and I can't really feel what the scalar field is.

Rocket Man
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2 Answers2

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Higgs boson is a scalar field. There are three types of fields in our universe, on which all the other fields fit in: we have spinors, and we have vectors. Spinor accounts for all mass: electrons, quarks, neutrinos, etc. While vectors account for light. Then we have 2-tensor fields for gravitation fields, and we have scalars.

What is the arranging pattern that organizes them together? Each of these type of fields can be conceived as array of numbers associated to each point of space-time. The defining property of each of them is the specific way in which those numbers 'rotate' when you rotate the frame of reference where you are measuring them.

Scalars and Vectors are the easier to grasp: a scalar is a single number, that keeps constant in all directions and all frames. Vectors are essentially arrows, so their components rotate exactly like what you would expect for an arrow, but with the caveat that arrow are 4D (with time component)

Spinors are two complex components, which rotate under rotations of SU(2) group, which is almost the same as O(3), except the subtle issue of the double covering. Suffice to say that it takes two whole rotations on SU(2) to get a whole rotation on O(3)

In gravitational theory, scalars have a stress tensor that behaves as a cosmological constant - a positive energy density of the scalar expands space-time proportionally. This is why they are widespread in the literature when considering inflation scenarios

lurscher
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  • Your response was helpful, but I am left wondering about the vacuum energy due to each of these scenarios. I understand that there is an vacuum energy density due to the cosmological constant because it is innate in the space itself. But is there a vacuum energy density due to the quintessence scalar field? – Rocket Man Jun 28 '13 at 18:27
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    I'm not sure if it is appropiate to talk of 'vacuum' states of quintessence, as it is a classical field as it has been proposed. It has a positive definite energy density as any other scalar field, and at the current time it has negative pressure, which is what makes it to expand vacuum spacetime. – lurscher Jun 28 '13 at 20:43
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    k-essense, on the other hand, is a weird thing that doesn't fit as your usual scalar field, because it has a kinetic energy term that is negative. And to my knowledge, there is no physical motivation for such a thing, even as a low-energy limit of topological leftovers from Planck epoch – lurscher Jun 28 '13 at 20:45
  • @lurscher $\mbox{SL}(2,\mathbb{C})$ is a double covering of $\mbox{SU}(2)$, and we have an isomorphism $(\mbox{SU}(2) \times \mbox{SU}(2)) / \mathbb{Z_2} \simeq \mbox{S0}(1,3)$. This is not the same as $\mbox{O}(3)$. – Flint72 Apr 14 '14 at 19:38
  • @Flint72 ${\rm SU}(2)$ is simply connected and therefore is its own universal cover. In fact this holds for all ${\rm SU}(N)$ groups, which are all simply connected. In turn ${\rm SU}(2)$ is the universal cover of ${\rm SO}(3)$, that being why its algebra ${\frak su}(2)$ coincides with the angular momentum algebra. What is true about ${\rm SL}(2,\mathbb{C})$ on the other hand is that it is the universal cover of the Lorentz group ${\rm SO}(1,3)$. – Gold May 25 '21 at 01:24
  • Finally the precise relation between ${\rm SO}(1,3)$ and ${\rm SU}(2)$ is that the complexified Lie algebra of the former is equal to a direct sum of two copies of the complexified Lie algebra of the later. – Gold May 25 '21 at 01:24
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Scalar field - something that takes a scalar value at every point in "space" (which is not necessarily physical space). Some familiar examples are the temperature and Newton's gravitational potential, but note these are not fundamental scalar fields since the temperature is a statistical aggregate and Newton's gravitational theory has of course been superseded by General Relativity.

Compare vector field, which is something that takes a vector value at every point in "space". Examples are the electric field & local wind speed.

Scalar fields do not have to be in physical space. For example $\sin{xy}$ defines a 2D scalar field in an infinite plane ("space") - the field is scalar because at every point it takes a single value while having no direction, 2D because there are two coordinates ($x$ and $y$), and infinite because each of the two coordinates can take on any value from $-\infty$ to $+ \infty$.

Allure
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