Operations regarding tensor-product states

In summary, the basis vectors for a two-particle state can be written as ##|\mu_i \rangle |\nu_j \rangle## for orthonormal vectors ##|\mu_i \rangle, |\nu_j \rangle## spanning their single-particle Hilbert spaces, and the inner product of basis vectors can be written as ##\langle \mu_{i'}|\langle \nu_{j'}|\mu_i \rangle |\nu_j \rangle = \delta_{i,i'} \delta_{j,j'}##. However, if kets with continuous degrees of freedom are involved
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Confusion regarding the inner product of tensor-product states
I have a question on tensor-product states that I'd like to ask, thanks in advance!

1. The basis vector of a two-particle state can be written as ##|\mu_i \rangle |\nu_j \rangle## for orthonormal vectors ##|\mu_i \rangle, |\nu_j \rangle## spanning their single-particle Hilbert spaces. The inner product of basis vectors can be written as ##\langle \mu_{i'}|\langle \nu_{j'}|\mu_i \rangle |\nu_j \rangle = \delta_{i,i'} \delta_{j,j'}##

Does this hold for kets with continuous degrees of freedom? Eg momentum ##\langle p',q' |p,q \rangle = \delta(p-p') \delta(q-q')##. After reading Blundell's book on QFT however (screenshot provided below), I attempted to use creation and annihilation operators to obtain the above identity, but what I kept obtaining was ##\langle p',q' |p,q \rangle = \langle 0|\hat{a_{p'}}\hat{a_{q'}}\hat{a_{p}}^\dagger\hat{a_{q}}^\dagger|0\rangle = \delta(p-p') \delta(q-q') + \delta(q-p')\delta(p-q')##.
Screenshot 2019-06-19 at 5.44.00 AM.png
 
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It's all consistent! The point is that in your example from the book you deal with identical particles, and the QFT formalism includes the appropriate symmetrization (bosons) or antisymmetrization (fermions), i.e., you do not deal with the full product space but only with the totally symmetrized/antisymmetrized Fock states for bosons and fermions. A convenient basis for the Fock space are the occupation-number states, which is very natural, because you cannot know in any way which particle is which due to the indistinguishability of identical particles (with identical I mean that the particles carry exactly the same instrinsic quantum numbers, mass and various conserved charges of the Standard Model).

What's wrong in the book is to omit the spin indices. Only for scalar bosons there are of course none. Fermions necessarily have half-integer spin and thus you must include the spin degree of freedom in you single-particle basis.
 
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Ah yes, I completely forgot about the need to satisfy exchange symmetry, thanks for pointing that out!
 

Related to Operations regarding tensor-product states

1. What are tensor-product states?

Tensor-product states are quantum states that are formed by taking the tensor product of two or more individual quantum states. This means that the state of the whole system is a combination of the individual states of each subsystem.

2. How are tensor-product states used in quantum information processing?

Tensor-product states are used in quantum information processing to represent the state of a multi-qubit system. They are also used in quantum algorithms, such as the quantum Fourier transform, and in quantum error correction codes.

3. What are the properties of tensor-product states?

Tensor-product states have the property of being separable, meaning that they can be written as a product of individual states. They also have the property of being entangled, meaning that the state of the whole system cannot be described by the states of the individual subsystems.

4. How are tensor-product states different from entangled states?

Tensor-product states are a special case of entangled states, where the state of the whole system can be written as a product of individual states. In contrast, general entangled states cannot be written in this way and are more complex to describe.

5. What are some real-world applications of tensor-product states?

Tensor-product states have applications in quantum communication, quantum cryptography, and quantum simulation. They are also used in quantum computing algorithms to perform tasks such as factoring large numbers and searching databases.

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