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A definition of a scalar like

A scalar is a one-component quantity that is invariant under rotations of the coordinate system (see http://mathworld.wolfram.com/Scalar.html)

seems to exclude complex numbers from being scalars, because complex numbers are generally expressed as two-component vector-like quantities. However in physics we have things like complex scalar fields. In what sense can complex numbers be scalars in physics? Does it mean, that a scalar is defined in physics simply as any quantity (independent of the number of components), that is invariant under space-time transformations (translations, rotations and Lorentz boosts)?

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    $\begingroup$ "complex numbers are generally expressed as two-component vector-like quantities" - you can express them in terms of two real components, just like you can express a real number in terms of an integer and fractional component, or you can express a rational in terms of a numerator and a denominator. What counts as a "one-component quantity" is highly context-dependent. $\endgroup$ – user2357112 supports Monica Sep 12 '16 at 20:15
  • $\begingroup$ Under rotations of which coordinate system? Perhaps not the one that has 1 and i as axes. $\endgroup$ – user253751 Sep 13 '16 at 4:40
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A scalar is a one-component quantity that is invariant under rotations of the coordinate system

OK, but what then do you mean by "rotation"?

See, a scalar in the sense as defined in your quote is not just "a scalar", period. You can only have a scalar with respect to some particular rotation operation. The same quantity can be a scalar with respect to one kind of rotation and a vector or tensor with respect to another.

It's true that there is a rotation group (a $U(1)$ group) which acts on the complex plane and turns one complex number into another. But that's not the type of rotation group physicists use. We use rotations that turn physical directions into one another (the traditional $SO(3)$ rotation), or that turn worldline directions into one another (the Lorentz group), or that turn spin states into one another (any $SU(2)$ spin group), or color states (the $SU(3)$ group used in QCD), or so on. None of these rotations affects a plain old complex number, because a plain old complex number doesn't have any physical meaning attached that would cause it to change under any physical rotation operation.

This has implications for what counts as a "component". As user2357112 mentioned in the comments, it depends on context: for example, you can treat a complex number as a two-component vector, or you could have a vector with complex coefficients (as in quantum mechanics), in which case each complex number is only one component. In fact, there are even situations where an entire matrix can be a component, such as the Pauli vector.

The point is that you shouldn't assume a component has to be a real number, or even any sort of number. It probably makes more sense to define a component in terms of rotations (since in math the whole idea of components comes from vector spaces, so we might as well do the analogous thing in physics). I'm not going to suggest any sort of rigorous definition here, but a sensible one would capture the idea that components of a vector "trade off" among each other under a rotation, and if some mathematical object isn't affected by a certain rotation, then the whole object (whether it's number, vector, tensor, whatever) deserves to be considered one component (and thus a scalar) with respect to that rotation.

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    $\begingroup$ But isn't $U(1)$ an isomorphism to $SO(2)$? $\endgroup$ – asmaier Sep 12 '16 at 8:00
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    $\begingroup$ Sure. You could call that rotation group $SO(2)$ if you prefer. It doesn't change anything about my answer, though. $\endgroup$ – David Z Sep 12 '16 at 9:16
  • $\begingroup$ This addresses the "invariant under rotations" part, but it doesn't say anything about the "one-component" part. It seems to me like the question was more about the components than the rotations. $\endgroup$ – user2357112 supports Monica Sep 12 '16 at 20:12
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    $\begingroup$ @user2357112 I thought the opposite. Actually, I would argue that part of the definition is unnecessary: a scalar (in the physics sense) need not be a one-component object, in the sense of each component being a single number. You could have a vector which is not affected by a particular rotation group, and that vector is a scalar for purposes of that rotation group. (IOW it falls in the trivial representation of that rotation group.) I suppose you could say it's not a scalar since it has multiple components, but then it's not a vector under that group either. $\endgroup$ – David Z Sep 13 '16 at 3:58
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    $\begingroup$ @asmaier In that case the complex number has some specific physical meaning attached that specifies how it transforms under rotations. It's no longer just a complex number. $\endgroup$ – David Z Sep 13 '16 at 4:01
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Although this is kind of trivial, a complex number, as a member of a field can be a scalar that acts by commutative multiplication on a vector space, the latter, through scaling, being the fundamental manifestation of the the notion of linearity. See the definition of a vector space for more details.

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To amplify WetSavannaAnimal's answer a bit, a mathematician defines a vector space (loosely) as a set of things that behave like little arrows when added together or multiplied by a scalar (aka number). They need not be little arrows. E.G. The set of all functions $y = ax^2 + bx + c$ is a 3D vector space.

An n dimensional vector can always be represented by n numbers, which is equivalent to a point in an n dimensional physical space, or a little arrow from the origin to that point. This is the sense in which a vector can be described by a magnitude and direction.

For the most familiar vector spaces, the numbers are real. But it is possible for them to be complex as well. E.G. the functions above could be defined over the complex plane. It would still be a 3D vector space. Even though $a$, $b$, and $c$ would be complex numbers, there are 3 of them.

This stretches the idea of an arrow in a physical space a bit. But then, so does a 4D or 17D vector. The point is that a scalar is the number that can multiply a vector without changing its direction.

To a physicist, a vector has to have another property. It must have a physically meaningful magnitude that does not change when you rotate the coordinate system. To a physicist, force is a vector, but a point in a thermodynamic phase space is not. To a physicist, 4D space-time is a vector space where the magnitude is the interval and the coordinate rotations are boosts.

Physicists are a bit sloppy on this point. To a mathematician, the idea of magnitude is captured by the definition of a norm. To a mathematician, 4D space-time is not a normed vector space because a norm must never be negative.

Getting back to the point, a second meaning of scalar is a physically meaningful value that is invariant under coordinate rotations. The magnitude of a vector is a scalar. Likewise, magnitudes of higher rank tensors are scalars.

In this sense, scalars are usually real numbers. Quantum mechanics has complex valued wave functions. But the physically meaningful magnitudes are real.

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Complex numbers are usually visualized as a "two-component vector-like quantity". However, this is just a visualization tool, and the real+imaginary axes of the Argand plane do not correspond to any physical directions. Complex numbers do not change under $SO(3)$ rotations of space or Lorentz boosts, which is why they are scalars.

If you think complex numbers are fundamentally linked to points on a 2-D surface, you might be interested in their history. Many important theorems about complex numbers were developed in the 18-th century, including de Moivre's formula and Euler's formula. All of these were based on the algebraic definition $i^2 = -1$, without any geometric identification/visualization of complex numbers as points in a complex plane. It was only in the 19-th century that the complex plane was born as a concept.

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  • $\begingroup$ The fact that a complex number and a 2-D vector both can be represented by a pair of real numbers, and both can be identified with a point in a plane is a shallow, misleading coincidence. The algebra of complex numbers is completely different from the algebra of vectors, and it's the two different systems of algebraic rules that make them both interesting. $\endgroup$ – Solomon Slow Sep 12 '16 at 13:45
  • $\begingroup$ But sometimes the real and imaginary axes do correspond to physical directions: en.wikipedia.org/wiki/… $\endgroup$ – asmaier Sep 12 '16 at 21:17
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What you've quoted is a definition of a "scalar" in some physical/mathematical context.

The term "scalar" comes from the Latin word scala meaning ladder; and multiplying a vector quantity by a scalar has the effect of scaling its magnitude without affecting its orientation. Hence the name "scalar". But over the years, "scalar" has gradually been bastardized by mathematicians to even refer to complex quantities "scaling" some other abstract mathematical quantity via multiplication. Despite the fact that originally, multiplying a vector by a complex quantity had the effect of both scaling and rotating a vector!

So a complex number can be a scalar today when it's used to "scale" another mathematical abstract quantity via the unary operation we call multiplication. But in a way that wasn't originally intended through the definition of a "scalar".

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