1

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?

J. W. Tanner
  • 63,683
  • 4
  • 43
  • 88
Kieran Mullen
  • 190
  • 6
    The third definition is the one that I use. What makes something an "allowed element of a field" doesn't make sense to me... all elements of our field are scalars and the field itself is the set of scalars, referred to as the scalar field. As for which fields are allowable as a scalar field, the answer is any field so long as scalar addition and multiplication is defined over your vector space. As for what makes a particular set of elements a field in the first place, this is well documented already in any introductory book. – JMoravitz Sep 18 '19 at 17:15
  • 1
    @JMoravitz I believe that comment can stand on its own as an answer :) –  Sep 18 '19 at 17:15
  • @JMoravitz I'd tweak this to say only that scalar multiplication needs to be defined on the vector space in such a way that it distributes over vector addition. – Robert Shore Sep 18 '19 at 17:18
  • I just want to add that your first statement is somewhat misleading 'In physics a scalar ...' Sure, in high-school level physics, it is usually intuitive to explain it that way. In the same fashion a vector is defined as a thing with direction in basic algebra classes. But that doesn't imply a Mathematician defines a vector that way, nor does it imply that in Physics an scalar is considered to be a quantity wholly defined by a magnitude and no direction. – caverac Sep 18 '19 at 17:47
  • 1
    Do we really need a rigorous general definition of this term? To me this term seems to me to be one for which a general definition (for which there would for sure be exceptions wrt its use in certain contexts) would not be of much value. – Winther Sep 18 '19 at 17:57
  • 2
    Possible duplicate of Scalar-valued function definition – Xander Henderson Sep 19 '19 at 12:15
  • Also related: https://math.stackexchange.com/questions/1834686/why-do-we-use-the-word-scalar-and-not-number-in-linear-algebra – Xander Henderson Sep 19 '19 at 12:16

3 Answers3

3

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.

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.

Community
  • 1
Noah Schweber
  • 260,658
  • So by this definition the integers are not a set of scalars, because they don't have a multiplicative inverse? – Kieran Mullen Sep 18 '19 at 17:54
  • @KieranMullen Well, the integers with the usual arithmetic structure, but yes. – Noah Schweber Sep 18 '19 at 18:02
  • The above answer has content answer is helpful - if your elements cannot be used to form a field then they are not scalars (under this definition). I disagree strongly with the statement that the question is an empty one. A scalar is a quantity that can support the structure of a field. Some quantities cannot and you can say that they are not scalars. – Kieran Mullen Sep 18 '19 at 22:36
  • @KieranMullen It still doesn't make sense to ask whether an individual thing can be a scalar - any individual thing can be an element of a field. If $a$ is any object at all, pick any other object $b\not=a$; then ${a,b}$ is the underlying set of a field, indeed two different fields. So anything whatsoever can be a scalar - there is indeed nothing to say here. As to what sets of things can be the underlying set of a field, the only restriction is size: $X$ is the underlying set of some field iff $X$ is either infinite or of prime power size $\ge 2$. – Noah Schweber Sep 18 '19 at 23:05
0

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.

Patrick Nicodemus
  • 1,342
  • If you cannot say what a scalar is can you tell when something is not a scalar? – Kieran Mullen Sep 18 '19 at 22:41
  • I'm afraid I don't understand your comment. I am telling you that in the context of a fixed pair $(F,V)$, where $F$ is a field and $V$ is a vector, I define a scalar to be an element of the field. You asked if the notion of scalar makes sense outside of the notion of a field (or ring): arguably, the answer is no. – Patrick Nicodemus Sep 18 '19 at 23:57
0

I believe the confusion is caused by the conflicting uses of the term in mathematics and physics.

In mathematics, a scalar is an element of the field over which the vector space is defined. They act on vectors through scalar multiplication.

In physics, a scalar is a quantity described by a single number and has an associated unit - it represents a real world entity. Unlike elements of a field, these types of scalars cannot be multiplied together e.g. multiplying two different masses does not yield another mass quantity. They can only be added together or multiplied by a dimensionless number i.e an element of a field, usually the real numbers. In this sense, a scalar could thought of as almost being one dimensional vector space. ("Almost" since you cannot have negative mass etc...)

dk30
  • 53