Field theory in non-relativistic QM

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Hello fellow physicists!
I'm currently trying to learn some QFT and the reader gives an introduction by expressing the non-relativistic hamiltonian with integral and creation, destruction operators.

Later he writes:

|Psi, t > = \int d3x1...3xn Psi(x1, ..., xn; t) a+(x1) ... |0>

And i´m not really sure how to think about this. Isn't |Psi, t > the wavevector of Psi(x1, ..., xn; t) ?
Please give me some handwaving or mental words about why this is true.. :confused:
 

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A. Neumaier
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|Psi, t > = \int d3x1...3xn Psi(x1, ..., xn; t) a+(x1) ... |0>

Isn't |Psi, t > the wave vector of Psi(x1, ..., xn; t) ?:
[tex]|x_1,...,x_n>:=a^*(x_1)...a^*(x_n)|0>[/tex] is a basis vector,
[tex]\Psi(x_1,...,x_n,t) [/tex] is the wave function, and
[tex]\Psi(t)=|\Psi,t>[/tex] is the wave vector.

In first quantized notation, the wave function is regarded as a wave vector in a function space. Then [tex]|x_1,...x_n>:=\delta_{x_1}\otimes...\delta_{x_n}[/tex], where [tex]\delta_z[/tex] is the delta function with support at z. Then your formula reduces to a trivial identity.

This is a functional version of the representation of a vector v in 3-space as
[tex] v=(v_1,v_2,v_3)^T = v_1|1>+v_2|2>+v_3|3>,[/tex]
where |k> is the k-th unit vector.
 

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