What is the definition of a scalar?

Question

In physics a scalar is usually defined as a quantity wholly defined by a magnitude and no direction. This is not a great definition, since a complex number is not a scalar under that definition.

A second definition of a scalar is a quantity that transforms as a scalar (e.g. is unchanged) under a change of coordinates. This allows for pseudo scalars, and is useful in teaching vectors and tensors to physics graduate students. However, I suspect that a mathematician would balk at this definition since it relies on coordinates.

A third definition is that "a scalar is an element of a field which is used to define a vector space." However this has no content unless I am told what makes something an allowed element of a field. Must it have certain properties like a commutative, closed, binary operation with an identity? Does the concept of a scalar exist independent of the concept of a field?

Answer 1

I believe that this question is similar to asking "what is a number." In different contexts one makes use of the counting numbers, the integers, the real numbers, the complex numbers, elements of the algebraic closure of Q in C, and so on. For a discussion about a specific problem to be productive, one must specify, if only for sake of convention, what is meant by a number, or what the set of numbers is that is being considered.

Therefore, in my opinion, if two physicists are speaking, and one says "Take lambda to be a scalar", we should understand this as follows: there is some ordered pair (F, V) that has gone unspoken between them; where F is a field, and V is an Abelian group, and F acts on V in a way satisfying the vector space axioms. The precise choice of F and V may be irrelevant, if they are merely discussing abstract linear algebra, or it may be very important that F is some specific field like complex numbers, and V is a specific space like a signal space, if a concrete problem is at hand. In any case, a scalar is an element of the field F being implicitly or explicitly discussed.

Answer 2

This is definitely a situation where added generality yields more clarity.

Although in particular contexts only some vector spaces may be of interest, a vector space in general is just the following:

A vector space is a triple $$(\mathcal{V},\mathcal{F},\cdot),$$ where

• $\mathcal{F}=(F; +_F,\times_F,0_F,1_F)$ is a field,

• $\mathcal{V}=(V; +_V)$ is a group, and

• $\cdot$ is a function from $F\times V$ to $V$ satisfying the rules (for all $a,b\in F$ and $u,v\in V$)

  • $1_F\cdot v=v$

  • $(a+_Fb)\cdot v=(a\cdot v)+_V(b\cdot v),$

  • $(a\times_Fb)\cdot v=a\cdot(b\cdot v)$, and

  • $a\cdot (u+_Vv)=(a\cdot u)+_V(a\cdot v)$.

Any triple of this type is a vector space. In particular, any field can serve as the field of scalars of a vector space, and it's a good exercise to show that a field $\mathcal{F}$ is a vector space over itself in a natural way: take $\mathcal{V}$ to be the underlying additive group of $\mathcal{F}$, and take the scalar multiplication to just be the multiplication of $\mathcal{F}$.

When we're being more specific it should be clear from context what sorts of vector spaces are being considered. Often, for example, we're only interested in the case when $\mathcal{F}=\mathbb{C}$, or $\mathcal{F}=\mathbb{R}$. However, in general there's absolutely no restriction on what fields can serve as the field of scalars of a vector space.

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There's a slight slipperiness here, though: what exactly are elements of fields? Sure, any field can be the field of scalars, but what sorts of things can be in fields to begin with? This gets to a "structuralist" aspect of mathematics: we don't care about what the underlying set of a structure - such as a field - actually is, but rather the behavior of that set together with the additional functions/relations/whatever. So anything whatsoever can be an element of the underlying set of a field, and so from a purely mathematical standpoint the question "what is an individual scalar?" isn't really meaningful. Of course, in particular situations we're often only interested in limited examples, and then we can say more - e.g. maybe we only care about subfields of $\mathbb{C}$, in which case we can get away with saying "all vectors are complex numbers," even though strictly speaking that's a bit bunk.