14.11.2 Tensor Operator Product Domain Tensor Space

Explore how tensor operator product domains define tensor spaces through algebraic structures and their operational interactions.

Tensor Operator Product Domain Tensor Space is the tensor product space that supplies the input to a combined operator built from individual factor operators, formed as the tensor product of the domains of each factor operator in the same order as the corresponding factors appear in the original tensor product.

Formation of the Domain

Domain of Each Factor Operator

Each factor operator entering the construction has its own domain, a vector space on which that single operator is defined, independent of the domains of every other factor operator involved.

T_{1} : V_{1} \to W_{1}

T_{2} : V_{2} \to W_{2}

The Combined Domain

The combined operator receives as input a vector from the tensor product of the individual domains, so the domain tensor space is fixed entirely by the domains of the factor operators, before anything is known about their codomains.

T_{1} \otimes T_{2} : V_{1} \otimes V_{2} \to W_{1} \otimes W_{2}

Structural Diagram of the Domain

Assembling the Domain From Individual Factor Domains

The diagram below shows two individual factor domains combining into a single domain tensor space that serves as the input to the combined operator.

Dimension of the Domain Tensor Space

Multiplicative Dimension Relation

The dimension of the domain tensor space equals the product of the dimensions of the individual domain spaces of the factor operators, so a small increase in either individual domain dimension produces a proportionally larger increase in the domain tensor space dimension.

\dim (V_{1} \otimes V_{2}) = \dim (V_{1}) \times \dim (V_{2})

Independence From the Codomain Dimensions

The dimension of the domain tensor space is determined entirely by the domains of the factor operators and has no dependence on what the codomains of those operators happen to be, so factor operators with very different codomains can still share the same domain tensor space.

Basis Induced on the Domain

Basis Vectors of the Domain Tensor Space

Once a basis is fixed for each individual domain space, the domain tensor space inherits a basis consisting of all simple tensors formed by pairing one basis vector from each individual domain, enumerated according to a fixed ordering convention.

Coordinates of an Input Vector

Any input vector to the combined operator is expressed, relative to this induced basis, as a coordinate column whose length equals the dimension of the domain tensor space, ready to be multiplied by the matrix representation of the combined operator.

Special Cases of the Domain

Operators Sharing a Common Domain

When several factor operators are all defined on the same underlying vector space, the domain tensor space is formed by repeating that same space as many times as there are factor operators, one copy for each position in the tensor product.

A Single Factor Operator With Identity Elsewhere

When only one factor operator is nontrivial and every other position uses the identity operator, the domain tensor space is still the full tensor product of all the factor domains, even though the combined operator changes only the component corresponding to the nontrivial factor.

Extension to Several Factors

Domain for a Product of Many Factor Operators

When the combined operator is built from three or more factor operators, its domain tensor space is the tensor product of all the individual domain spaces, taken in the same order as the corresponding factor operators appear in the construction.

Consistency With Grouping of Factors

Regardless of how the factor domains are grouped when forming the repeated tensor product, the resulting domain tensor space is the same, since the tensor product of the individual domains does not depend on the order in which the factors are associated together.