The physical meaning of the tensor product

The concept of tensor products of vector spaces is required for the description of General Relativity. For example, the metric tensor $ \mathbf{g} $ of a pseudo-Riemannian manifold $ (M,\mathbf{g}) $ can be viewed as a ‘disjoint collection’ of tensor products. In terms of a local-coordinate system $ (x^{1},\ldots,x^{n}) $, we can express $ \mathbf{g} $ as $ g_{ij} \cdot d{x^{i}} \otimes d{x^{j}} $, where $ d{x^{i}} $ and $ d{x^{j}} $ are differential $ 1 $-forms and $ g_{ij} $ is just a scalar coefficient. At each $ p \in M $, what $ \mathbf{g} $ does is to take two tangent vectors at $ p $ as input and produce a scalar in $ \mathbb{R} $ as output, all in a linear fashion. In other words, at each point $ p $, we can view $ \mathbf{g} = g_{ij} \cdot d{x^{i}} \otimes d{x^{j}} $ as a bilinear mapping from $ {T_{p}}(M) \times {T_{p}}(M) $ to $ \mathbb{R} $, where $ {T_{p}}(M) $ denotes the tangent space at $ p $.

The metric tensor is the very object that encodes geometrical information about the manifold $ M $. All other useful quantities are constructed from it, such as the Riemann curvature tensor and the Ricci curvature tensor (of course, we still need something called an ‘affine connection’ in order to define these tensors). In General Relativity, the metric tensor encodes the curvature of spacetime, which can and does affect the performance of high-precision instrumentation such as the Global Positioning System (GPS).

For vector spaces over a field $ \mathbb{K} $, tensor products are usually defined in terms of multilinear mappings. You may have seen the more abstract definition using a system of generators, but it can be shown that these two definitions are the same. From the categorical point of view, both constructions satisfy the same universal property (this property is explicated in the Wikipedia article on tensor products), so they must be isomorphic in the category of $ \mathbb{K} $-vector spaces.

You need to understand that tensor products are algebraic objects, while tensor fields are geometrical objects. Even more fundamental than the concept of a tensor field is that of a tensor bundle. You can think of a tensor bundle as a collection of tensor products attached to individual points of a manifold in a consistent manner (by ‘consistent’, I mean that the axioms of a vector bundle must be satisfied). How we relate the tensor product attached to a point to the tensor product attached to another point is what the subject of differential geometry is all about. On the other hand, what we do with the tensor product attached to a fixed single point is pure algebra.

Any discussion of spacetime on a global scale requires General Relativity, which takes into account the curvature of spacetime. However, in a small neighborhood of any fixed point in spacetime, one always has what is called a ‘local inertial frame’. In such a frame, gravitational effects can be transformed away, in which case, General Relativity simply reduces to Special Relativity, which some mathematicians view as nothing more than just advanced linear algebra. This illustration reinforces the idea that ‘global is geometry but local is algebra’.

This is a really insightful question! You're pointing out that, in quantum mechanics, there are two different notions of "tensor product":

• The composition of physical systems (a physical notion).
• The tensor product of Hilbert spaces (a mathematical notion).

You say you can't see how these two notions are equivalent, and you're right! They aren't.

In different kinds of physical theories, the composition of physical systems is represented mathematically in different ways. For example:

• In statistical mechanics, each type of physical system is described by a measurable space (a space that can carry probability distributions), and the composition of systems is described by the Cartesian product of measurable spaces.
• In classical mechanics, each type of physical system is described by a symplectic manifold (a space where Hamilton's equations make sense), and composition is described by the symplectic product of symplectic manifolds.

There's a mathematical idealization that perfectly captures the physical idea of composition of systems: the "monoidal tensor product" in a monoidal category. The tensor product of vector spaces, the Cartesian product of measurable spaces, and the symplectic product of symplectic manifolds are all examples of monoidal tensor products.

To learn more about monoidal categories, and their deep relationship with physics, I recommend Bob Coecke's "Introducing categories to the practicing physicist." You might also want to check out a classic but more background-intensive article by John Baez, called "Quantum Quandaries: A Category-Theoretic Perspective."

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p.s.

As I've been saying, the use of the tensor product of Hilbert spaces to represent the composition of physical systems is not requried—in fact, even the use of Hilbert spaces to represent types of physical systems is not required! So, why do we always use the mathematical machinery of Hilbert spaces, linear maps, and tensor products to build our quantum theories?

If you learn enough category theory, you can find a really cool answer in Chris Heunen's thesis, "Categorical quantum models and logics." Heunen shows that the bundle of mathematical machinery I mentioned above (usually called "the category of Hilbert spaces," or Hilb for short) is sophisticated enough to build any theory that satisfies certain conditions (see Definitions 3.7.1 and 3.6.1 and Theorem 3.7.8). The conditions look pretty technical, but many of them—maybe all of them!—can be thought of in an intuitive physical way. Roughly speaking, I'm pretty sure that...

• "It has a dagger" means your theory has a notion of time-reversal. The adjective "dagger" that shows up in the other conditions means "compatible with time reversal."
• "It is symmetric dagger monoidal" means your theory has a notion of composition of physical systems.
• "It has finite dagger biproducts" means that if your theory can describe systems of type proton and systems of type neutron, it can also describe systems of type proton or neutron, but I don't know which, and that you can always do an experiment to determine whether a system of this type is a proton or a neutron.
• "Every dagger mono is a dagger kernel" means that if the type car has a subtype awesome car, there's also a subtype not-awesome car.
• "It has finite dagger equalisers" means that for any two experiments that can be done on the same type of system, there's a subtype describing systems for which the outcomes of the experiments are always the same.