What is *physical meaning*?

Question

What do we mean when we talk about physical meaning of a quantity, an equation, theory, etc.? Should the physical meaning touch on the relation between the math and the real world? Or does it have more to do with how the equation/theory is used by physicists?

Background
For the immediate background that prompted me to ask this question see the discussion that followed answers to this question.

This forum contains nearly 3000 questions of the type What is the physical meaning of X... but do we know what we are asking?

Opinion
I think the question is important, because it defines the special place of physics among other disciplines. When we ask about a physical meaning of something we really ask how this something is related to the real world, as opposed to purely mathematical reasoning. Mathematicians and biologists do not question mathematical or biological meaning of their objects of study, since it is obvious. Yet, physicists must justify their calculations by basing them on the experimental data and making experimental predictions (as opposed to mathematicians). In the same time physicists cannot do experiments without developing complex mathematical models (unlike biologists or chemists - even though these are often more knowledgeable about complex statistical methods than an average physicist.)

Remark
As of 28/04/2025 the search in this community produces nearly 5000 hits on physical meaning. E.g.: What is the physical meaning of phase in a superconductor?.

Answer 1

Let me first ask you a question; what do you think I mean by $$\mathbf{F} = m \mathbf a$$ ?

From a mathematical point of view, the equation expresses the relationship between two vectors.

However, a physicist, when using mathematics to understand nature, makes mapping between physical concepts and mathematical objects. For example for the above case, there is a measurable quantity & a physical concept called force and we are denoting it by a mathematical object, namely a vector, so the mapping is

$$\text{Force (measurable quantity)} \to \vec{F} (\text{mathematical object}).$$

Now, coming back to your question, what do they mean by "physical meaning of a mathematical expression" is the inverse mapping of the above relations, i.e

$$ \vec{F} (\text{mathematical object}) \to \text{Force (measurable quantity)}$$

Answer 2

This is a deep question, with important implications for understanding the mathematical form of both relativity and quantum mechanics. A quantity is a numerical quantity, and an equation expresses a relationship between quantities. The question can be largely answered by describing what is a physical quantity. As Eddington put it

• “A physical quantity is defined by the series of operations and calculations of which it is the result.” (Eddington A.S., 1923, The Mathematical Theory of Relativity, 2nd ed., p. 3, CUP)

This is in stark contrast to the classical idea that physical quantities exist in nature, and that the measurement merely determines their value. Eddington was writing in the context of relativity, but in quantum mechanics Dirac wrote:

• “In the general case we cannot speak of an observable having a value for a particular state, but we can … speak of the probability of its having a specified value for the state, meaning the probability of this specified value being obtained when one makes a measurement of the observable.” (Dirac P.A.M., 1958, Quantum Mechanics, Clarendon Press, p.47)

Again it is seen that measured quantities are the product of measurement procedures, not physical priors in the structure of matter. The mathematical theories of physics largely describe relationships found in measurement, but they go outside of that also, because we develop mathematical structures which have no direct analogue in physical reality.

Much of the misunderstanding of quantum mechanics, and indeed the belief that quantum mechanics cannot be understood, is derived from the mistaken idea that mathematics describes reality. Actually mathematics is simply language, and like other languages it can describe that which does not exist in nature. Mathematics is governed by logic. It can be used to deduce relationships found in measurements using logical arguments containing logical steps which have nothing, directly, to do with physical reality.

Answer 3

I understand physical meaning as setting the context.

We can learn a lot about the maths of 2-spin particles, operators, probabilities and expected values for example. But without learning about Zeeman effect and Stern-Gerlach experiment, all the stuff seems coming from nowhere.

Answer 4

Many times physical meaning is just a sentence we use to cover our lack of a better mathematical understanding. However even in math one can ask about the deeper meaning of a certain result. So in some situation it's definitely a legitimate question.

I will answer by giving you an example. Let's suppose that we want to describe a quantum mechanical point particle with one degree of freedom (living on the real line). A mathematician that studied functional analysis will tell you that a state is an element of the Hilbert space $L^2(\mathbb{R})$. This is perfectly correct.

However it all makes sense physically. The reason is that Born rules tells us that for a particle described by wavefunction $\psi$ the probability of finding it in the set $\Omega$ is

$$ \int_\Omega dx |\psi(x)|^2 \tag{1} $$

Since the total probability must be one, we see that the wavefunction must be square-integrable. Moreover, since wavefuntions that differ on a set of measure zero give the same result for quantities such as (1), we realize that a quantum state is actually not a function but rather an equivalence class of functions that differ on set of measure zero. We just built, physically the mathematical space $L^2(\mathbb{R})$.

It is quite astonishing that the mathematical theory of $L^p(\mathbb{R})$ spaces has been put forward independentely (by Riesz) more or less (or a little bit earlier) in the same times as quantum mechanics was being developed.