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fa.useful evaluation – Quantum fields and innumerable tensor merchandise

For free quantum fields I cerebrate this loom can breathe handled utilizing this concept:

- John Baez, Irving Segal and Zhenfang Zhou,
*Introduction to Algebraic and Constructive Quantum Field Theory*, Section 4.5: Infinite merchandise of Hilbert areas, Princeton U. Press, 1992, pp. 125-130. Available in pdf and dvju format right here.

We narrate a well-behaved understanding of **grounded tensor product** for a presumably innumerable assortment of **grounded** Hilbert areas: that’s, Hilbert areas $(K_lambda)_{lambda in Lambda}$ geared up with unit vectors $z_lambda in K_lambda$. If every $K_lambda$ is separable and the index clique $Lambda$ is countable, this tensor product is separable!

This would not ameliorate you for a tensor product of *uncountably many* Hilbert areas, but it surely quiet helps you *a miniature* along with your query. There is a route to dwindle the Hilbert area of a free quantum bailiwick to an innumerable however countable tensor product of grounded Hilbert areas.

Namely:

When you will have a free bosonic quantum bailiwick, the single-particle Hilbert area $H$ is a countable direct sum of 1-dimensional areas $H_lambda$. Quantizing every $H_lambda$ is simply love quantizing a harmonic oscillator: the Fock area of $H_lambda$, say $K_lambda$, is a Hilbert area completion of the polynomial algebra on $H_lambda$.

Moreover, every $K_lambda$ is grounded: there’s an patent ‘vacuum vector’ $z_lambda in H_lambda$, particularly the component 1 within the polynomial algebra. And here is the very best sever: the Fock area of $H$, say $Okay$, is the grounded tensor product of the $K_lambda$:

$$ H = bigoplus_{lambda} H_lambda implies Okay = bigotimes_{lambda} K_lambda $$

the place, simply to emphasise, the tensor product right here is the *grounded* tensor product.

If we’re coping with a free quantum bailiwick on a spacetime $mathbb{R} occasions S$ the place the spatial manifold is compact, we will do the decomposition

$$ H = bigoplus_{lambda} H_lambda $$

utilizing momentum or vitality eigenstates, for the reason that Laplacian and different elliptic operators on $S$ may have discrete spectrum.

If we’re engaged on Minkowski spacetime, as you’re, this does not labor: your momentum $p$ takes a *continuum* of values. So you are making an attempt to jot down $H$ not as a direct sum however as a *direct integral* of 1-dimensional Hilbert areas.

So, what appear to breathe asking for is a generalization of the grounded tensor product to a kindly of ‘grounded steady tensor product’ operation that units up an affinity

direct sum : grounded tensor product :: direct integral : grounded steady tensor product

My hunch is that this could breathe doable. For one factor, physicists are implicitly utilizing a nonrigorous model of this thought of their every day labor on quantum bailiwick theory—as you identified. For one other, it is a kind of conditions the place the ultimate respond you are capturing for has been made rigorous, and also you’re simply searching for a fresh route of getting there.

However, I’m fortunate sufficient quick?witted that countable tensor merchandise of grounded Hilbert areas labor as they need to. In the bespeak, we employ them to analyze the query of when a linear symplectic transformation of $H$ can breathe quantized to secure a unitary operator on $Okay$.

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