# Substance like quanties and conserved quantities, Karlsruhe physics course

In the Karlsruhe physics course one defines the term "substance-like" quantity:

Let my cite the definition from a paper by Falk, Herrmann and Schmid:

"There is a class of physical quantities whose characteristics are especially easy to visualize: those extensive physical quantities to which a density can be assigned. These include electric charge, mass, amount of substance (number of particles), and others. Because of the fundamental role these quantities play throughout science and because such quantities can be distributed in and flow through space, we give them a designation of their own: substance-like."

Are there examples of extensive quantities, which are not substance-like? I think volume is one example, since it seems to make no sense to assign a density to it, are there others?

Now the authors write that a quantity can only be conserved if it is substance-like, let my cite this from an other publication:

F. Herrmann, writes: "It is important to make clear that the question of conservation or non-conservation only makes sense with substance-like quantities. Only in the context of substance-like quantities does it make sense to ask the question of whether they are conserved or not. The question makes no sense in the case of non-substance-like quantities such as field strength or temperature."

So my second question is: Why has a conserved quantity to be substance like? It would be great if one could give me a detailed explanation (or a counterexample if he thinks the statement is wrong).

Are there resources where the ideas cited above are introduced with some higher degree of detail and rigour?

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An example that might be useful in this context: Consider a density $\rho$ in the Lagrangian picture (as opposed to the Eulerian picture) of an incompressible fluid. Since $\rho$ does not scale with volume, it is an intensive quantity. Nevertheless, $\rho$ is a constant of motion. –  Qmechanic Apr 30 '11 at 18:25
After reading Luboš Motl's answer I thought that it is perhaps not clear to me what is meant by "conserved quantity". I think this means that it is a quantity whose "value" doesn't change with time when the system evolves determined by its "dynamics". But with this definition I see no problem to say that temperature or some other non-extensive quantity could be conserved. –  martin Apr 30 '11 at 18:28
@Qmechanic: So you would say that the cited assertion by F. Herrmann is not correct? –  martin Apr 30 '11 at 18:29

Concerning your first question, whether extensive quantities may fail to be substance-like, the answer is No. The very citation you quoted says that "substance-like" is a synonym that the authors chose for an "extensive" quantity.

Concerning your second question, yes, conserved quantities have to be substance-like i.e. extensive. It's because our world may be approximated by a field theory governed by the action which is an integral of the Lagrangian density over spacetime. $$S = \int d^d x \, {\mathcal L}$$ Conserved quantities are associated with symmetries and Noether's procedure applied to all theories described by such an action leads to a formula for the conserved quantity that is an integral over space, e.g. $$J = \int d^{d-1} x\, j_0$$ That means that $J$ is extensive.

Of course, if one could show that despite all the positive experiments of the last centuries, physics does not take place in space or it is not local, the argument above could break down.

I also want to mention that the substance-like character of the conserved quantities is kind of obvious. It just says that the conserved things must be "somewhere" in space. To mention a particular example of quantities that can't be conserved, it's the "intensive" quantities such as temperature and pressure.

Obviously, they can't be conserved because they're not even well-defined in a generic situation - different parts of the physical system have different pressures. If two similar objects have different temperatures, the "natural" other temperatures we can construct out of these two is (approximately) the average, rather than the sum, of the original two temperatures.

That means that addition - something we do with conserved quantities - isn't appropriate for intensive quantities. Instead, averaging is.

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Do you mean by "somewhere in space" that the physical quantity under consideration should be a map which domain is the space-time? However I could do the same thing for example with temperature (I assign a temperature value to every space-time point). –  martin Apr 30 '11 at 18:24
@martin No, the conserved quantity is the result of the integration, not the integrand. In the example given by Luboš, $J$ is the conserved quantity, not $j_0$. –  mmc Apr 30 '11 at 23:04