Why the tensor product (historical question)?

In summary, the founding fathers of QM knew that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces because the direct product excludes the possibility of entanglement and the Born rule would not work. While it is unclear what the historical reason for this is, it is mathematically necessary for the probabilistic interpretation to hold. Today, we can verify entanglement experimentally, but this was not possible when the EPR paper was published due to technological limitations.
  • #1
greypilgrim
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Hi.

Why did the founding fathers of QM know that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces and not a direct product, where no entanglement would emerge? I mean today we can verify entanglement experimentally, but this became technologically possible far later than for example the EPR paper.

My knowledge about functional analysis is pretty limited, is there a mathematical reason to exclude the direct product?
 
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  • #2
I'm not sure of the historical reason but it is at least sufficient for the Born rule to work. If ##P_1## and ##P_2## are transition probabilities for independent systems, the joint transition probability ##P## is the product $$ P = P_1 P_2 = | \langle \varphi_1 |\psi_1 \rangle |^2 |\langle \varphi_2 |\psi_2 \rangle |^2 = | ( \langle \varphi_1 | \otimes \langle \varphi_2 |)(|\psi_1 \rangle \otimes |\psi_2 \rangle |^2$$ If we use the direct product instead (which I think is the same as direct sum for finite-dimensional vector spaces), states of independent systems will be orthogonal and the probabilistic interpretation won't work.

See also this post https://www.physicsforums.com/threads/tensor-products-and-subsystems.360264/#post-2473399.
 

FAQ: Why the tensor product (historical question)?

1. What is the tensor product?

The tensor product is a mathematical operation that combines two mathematical objects (usually vectors or matrices) to create a new object that contains information from both original objects.

2. Why is the tensor product important?

The tensor product is important because it allows us to study the relationship between two or more mathematical objects, which can aid in solving complex problems and understanding abstract concepts in mathematics.

3. When was the concept of tensor product first introduced?

The concept of tensor product was first introduced in the mid-19th century by German mathematician Hermann Grassmann in his work on multilinear algebra. However, it was not until the early 20th century that the modern definition and notation of tensor product were established by mathematicians such as Élie Cartan and Hermann Weyl.

4. How is the tensor product different from other mathematical operations?

The tensor product differs from other mathematical operations such as addition and multiplication because it does not follow the usual rules of commutativity and distributivity. Additionally, the result of a tensor product is a new object with its own unique properties, rather than a simple combination of the original objects.

5. What are some applications of the tensor product?

The tensor product has many applications in mathematics, physics, and engineering. It is used in fields such as differential geometry, quantum mechanics, and signal processing to describe and analyze complex systems. It is also essential in the development of machine learning algorithms and data analysis techniques.

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