5.17.3 Tensor Product Complex Field Context

Tensor Product Complex Field Context is the specific instance of the tensor product construction in which the scalar field F is taken to be the field of complex numbers ℂ, making V and W complex vector spaces and V ⊗_ℂ W a complex vector space of dimension dim(V) · dim(W). This context introduces a subtlety absent from the real case: because ℂ carries a nontrivial field automorphism, complex conjugation, two genuinely different bilinear-type pairings can be formed from a pair of complex vector spaces, and care must be taken to specify which one is intended whenever complex tensor products are used to build inner products or physical quantum states.

The Ordinary Complex Tensor Product

Standard Bilinear Construction

The tensor product V ⊗_ℂ W is constructed exactly as in the general field case, with the free module F(V × W) taken over ℂ and homogeneity given, for c ∈ ℂ, by:

with c genuinely complex, not restricted to its real or imaginary parts.

Universal Property for Complex Bilinear Maps

As in the general case, V ⊗_ℂ W satisfies the universal property with respect to ℂ-bilinear maps: maps that are ℂ-linear in each argument separately, factoring uniquely through a ℂ-linear map from the tensor product, exactly as required by the field linearity requirement.

The Complication Introduced by Conjugation

Sesquilinear Forms Are Not Bilinear

Many important pairings on complex vector spaces, most notably Hermitian inner products, are conjugate-linear in one argument rather than linear:

Such sesquilinear forms are not bilinear and therefore do not factor through the ordinary tensor product V ⊗_ℂ W as constructed here.

The Conjugate Vector Space Resolution

To accommodate sesquilinear forms within the tensor product framework, one introduces the conjugate vector space V̄, which has the same underlying additive group as V but with scalar multiplication twisted by conjugation: c ·_V̄ v := c̄ v. A Hermitian form on V then corresponds to an ordinary bilinear map on V̄ × V, which does factor through the ordinary tensor product V̄ ⊗_ℂ V.

Diagram Distinguishing the Two Pairings

Applications Requiring Careful Field Context

Quantum Mechanics and Tensor Products of State Spaces

In quantum mechanics, the state space of a composite system is the ordinary complex tensor product V ⊗_ℂ W of the constituent state spaces, used to describe combined and entangled states, while the inner product used to compute probabilities relies on the conjugate-linear pairing on one factor, illustrating that both notions coexist but serve different roles within the same physical framework.

Complexification of Real Vector Spaces

A related but distinct operation, complexification, produces a complex vector space V_ℂ = V ⊗_ℝ ℂ from a real vector space V by extension of scalars, differing from the complex tensor product of two already-complex spaces, since complexification changes the base field of a single real space rather than combining two complex spaces.

Dimension and Basis Considerations

Complex Dimension Formula

As in the general field case, for finite-dimensional complex vector spaces V and W:

with basis tensors eᵢ ⊗ fⱼ formed from complex bases {eᵢ} and {fⱼ}.

Complex Matrices as Tensor Products

Under a choice of complex bases, V ⊗_ℂ W corresponds to the space of complex matrices Mat(dim V, dim W; ℂ), exactly analogous to the real case, connecting the abstract complex tensor product to ordinary complex linear algebra.

Broader Significance

Necessity of Specifying Which Pairing Is Meant

Because the complex field context admits two distinct pairing notions, bilinear and sesquilinear, any use of "tensor product" in a complex-vector-space setting should specify explicitly which construction is intended, particularly in physics literature where notational conventions sometimes elide this distinction.

Extension to Other Fields with Nontrivial Automorphisms

The same complication arising from complex conjugation reappears whenever the underlying field has a nontrivial automorphism, such as in the theory of Hermitian forms over quadratic field extensions, generalizing the conjugate-vector-space technique introduced here to a broader algebraic setting.