What is a physical quantity in science?

Question

The Wikipedia article on Physical Quantities, says that:

The meaning of the term physical quantity is generally well understood (everyone understands what is meant by the frequency of a periodic phenomenon, or the resistance of an electric wire). The term physical quantity does not imply a physically invariant quantity. Length for example is a physical quantity, yet it is variant under coordinate change in special and general relativity. The notion of physical quantities is so basic and intuitive in the realm of science, that it does not need to be explicitly spelled out or even mentioned. It is universally understood that scientists will (more often than not) deal with quantitative data, as opposed to qualitative data. Explicit mention and discussion of physical quantities is not part of any standard science program, and is more suited for a philosophy of science or philosophy program.

I'm interested in spelling out what a quantity is explicitly, and as suggested by Wiki this question is under the realm of philosophy, hence I ask this question here.

What are Physical quantities in science? How can we define them?

Answer 1

Physical quantity is something that one measures. In other words it is defined by the measurement procedure/protocol. Then one can form an abstract view of such a quantity, as something that can be measured via different procedures, and operate it as an independent category.

Wikipedia should be always taken with a grain of salt: what is described in the quoted fragment is a naïve view of physical quantity, as taught in low level physics courses. Scientists usually operate with clear definitions - either in terms of the measurement procedure, or in terms of other quantities (which are well defined) or in terms of properties that the quantity should possess.

Improperly defined quantities (sometimes intentionally) often generate scientific debates or sensational article titles, which tend to crop into popular media. E.g., one occasionally sees articles about violation of the laws in thermodynamics in molecules and nanosystems - but a qualified physicist knows that thermodynamic quantities are not defined for systems with few degrees of freedom (few atoms/molecules), and hence these claims cannot be taken seriously. A more technical example is spin current - measuring and calculating spin currents used to be a hot topic about a decade ago, but results were not consistent, because every researcher used its own measurement procedure, and these could never agree among themselves, because "spin current" cannot be made obey the current conservation law.

Answer 2

The definition of "physical quantity" depends on the context being used.

For classical physics, it is something that can be measured. And this requires a definition of "measure." To measure a thing is to establish a relationship with a measuring standard.

In previous eras, this standard might be one's own body. Or the body of some well known person, for example the king. This is probably the source of such standards as "one foot." One could establish the relationship by pacing-off a distance and counting the number of lengths of a foot in a distance to be measured. It would not be particularly accurate, and would be different for different foot sizes.

Such standards were formalized into, for example, the metric system. The standard meter and the standard kilogram and various other standards were created. The purpose for these was to establish a means by which different observers could agree on what the standard was, and so agree on what the measured value was. Modern standards are based on our understanding of such things as relativity. We thus have a standard second defined in terms of vibrations of a particular atom and the standard meter defined as the distance light travels in a certain time.

To establish the measurement relationship, we take the physical thing to be measured and find a means by which we can show it to be a multiple of the standard. To measure mass, for example, we establish that the test object is X times the mass of the standard kilogram. X can be any non-negative real number. There are a number of methods to accomplish this. For example, a balance scale can be used, based on the observation that balance beams follow the law of the lever. So if our test object balances at 1.7 times as far from the fulcrum as the standard kilogram, we conclude our test object is 1.7 kilograms.

Similar processes will apply for other standards. The standard of length will allow measurement of distances. And so on. In some cases we will use combinations of standards to establish combination physical measurements. Speed, for example, requires measuring a distance and a time. In each case we will establish a relationship with the measurement standard.

Two important features of physical quantities of this type are the following.

• They have physical units. These are established by the standard. Measurements of mass are in the unit of the standard, kilograms in the example.
• They are not absolutely accurate. That is, they are finitely accurate and have some degree of uncertainty.

An overlapping means of measurement is counting. This applies to collections of items that are similar in some way that is considered, in context, more important than any differences.

A common situation where this applies is counting oscillations in some such thing as a rotating wheel. If we assume a wheel is turning at a constant rate it makes sense to count the oscillations to determine the frequency. The time part of this measurement still gets measured as before, by establishing a relationship to a time standard. The oscillations are just counted. The sources of error here are the usual for the standard. Plus a new type with the possibility of being inaccurate in counting. The units for such are just "per second." (Or per unit time of your choice.)

Another case where counting applies is for items that are sufficiently standard that, for some purpose, we neglect the differences. We may buy bolts of some standard type by counting them. They will not necessarily be similar enough to be indistinguishable. But for the purposes of a building project they may be similar enough. Thus counting still may have some uncertainty associated. And the unit will be "one bolt" with uncertainty indicated by the variation in bolt mass, size, etc.

The meaning of "physical quantity" in quantum mechanics is hugely complicated and technical. And it generates discussion that can get warm. And it's quite annoying to try to explain it without using equations, which are $not enabled$ in this stack. Let me explicitly dodge doing that part.

Answer 3

The most common philsophical working framework for this kind of thing is known as operationalism, due to Percy Bridgman. Operationalism says that to define a physical quantity, you need to say what operations are required in order to measure it. In pure operationalism, one would completely eschew conceptual definitions. A good example is mass. Kids' science textbooks will often define mass as something like "a measure of the quantity of matter." But this is actually a pretty meaningless definition. Operationally, we would determine something's mass by, for example, putting it on a double-pan balance with a fixed standard object, or applying a standard force and measuring its acceleration.

Operationalism has various difficulties, such as the need to match up different orders of magnitude for a quantity that requires completely different measuring techniques in different regimes.

Many physical quantities include some ambiguity that makes them not directly measurable. For example, we can't measure absolute voltages, and we can't measure absolute phases of wavefunctions in quantum mechanics. However, we can measure differences in those quantities, and calculations that we write down in terms of their absolute versions can always be translated in an obvious way into calculations involving differences.

In reality, physicists actually work with a combination of operational and conceptual definitions. For example, relativists assign an entropy to a black hole and construct an entire system of black hole thermodynamics by analogy with the thermodynamics of ordinary matter. But with the present state of the art, there is no operational connection between black hole entropy and the entropy of a sample of air. The connection is purely conceptual, although the analogy leads physicists to believe that future developments will allow them to be better integrated.

Answer 4

Like many concepts, the term has evolved over time.

The earliest use of the term I have been able to find is in Euler's "Letters to a German Princess", where he tries to explain how is it geometers/natural philosophers (scientist-mathematicians) are able to adscribe mathematical quantities to physical entities, naturally the idea has predecessors, in Galileo for example, and perhaps Francis Bacon (personally, I have not read him, so I can't confirm). This idea is based on the division between the "mental" and the "material", a physical quantity being something material which a mental being could comprehend by adscribing a mathematical quantity.

As @Roger Vadim has mentioned, the particular definition of a particular physical quantity will depend on the "measure" to which it is subjected to. Although a kilo of flour would remain "a kilo of flour" now as it was 100 years ago, "kilogram" would mean conceptually different things (a priori). A good reference on the subject is Max Jammer's "Concepts of Mass in Contemporary Physics and Philosophy", which tracks how the word "mass" has changed through scientific development to mean related, but slightly distinct meanings.

What a "physical quantity" is, depends on what do you expect of the word "is". In practice, what a scientist would mean would be in the union of conceptual parts, a "generic" part, which they share in common with the broader scientific community, and another part related to their personal beliefs (in the ideal world of ideal scientists the latter is minimal). The personal part will vary according to what their particular ontological-epistemological stance is. In a rather (not completely absolute) strict sense, we can only "really" know that which we can measure, adscribing any other quality or attribute that we cannot confirm is just conjecture, which is not forbidden in scientific circles, but is looked-down upon the longer the chain becomes. Operationalism, as @cRdk9y36 points out, is both minimalist and pragmatic in its approach; not without its shortcomings, but still, I would say, the standard paradigm.

@Boba Fit points out the distinction in the ontology between Classical Physics and Quantum Physics, how you define "physical quantity" will vary whether you wish to have one definition which encompass both areas, or two definitions, each corresponding to one subject, where each would be related to the other by a correspondence (for example, Principle, and 2). I'll follow his lead by sidestepping the question of Quantum Realism. I must mention though, not all measurable quantities are necessarily "measured" by commensurablity (i.e.:That block over there is Nx(Q), because it measures N times the quantity Q), even in Classical Physics (dimensionless quantities, though commensurable, are not defined by their units, which they don't have)

(Most of this could be written as a series of comments under those cited, but I did not have have enough reputation, my apologies)