Abstract
This chapter is mainly based on the lectures presented by Mark Fannes at the COST Physics School on “New trends in many-particle quantum transport” of February 2015. These lecture notes were, in turn, based on several rather technical textbooks (Alicki and Fannes 2001; Benatti et al. 2010; Bratteli and Robinson 1987, 1997; Davidson 1996; Dereziński 2006; Evans and Kawahigashi 1998; Petz 1990) and we refer the interested reader to these works for more mathematical background on the results presented here. Our goal is to bring these results from the mathematical quantum physics community closer to the current mainstream quantum physics research.
More is different
Philip W. Anderson in (1972)
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- 1.
In a mathematical phrasing, one may think of the observable which describes the measurement.
- 2.
For single-particle systems there typically is no need to proceed to more abstract algebras of observables, since there usually the GNS construction (see Sect. 2.3.2) does lead to a unique Hilbert space.
- 3.
Classically this would be connected to an increase of the dimensions of phase-space.
- 4.
Because the group \(S_2\) contains only two elements, one of which is the identity, it is sufficient to focus on the action of U.
- 5.
In the next section, the Schur-Weyl duality is more generally formulated in Eq. (7.22).
- 6.
A total set in the context of topological vector spaces is a set, the linear span of which is dense in the full topological vector space. In a Hilbert space this becomes equivalent to stating that the only vector orthogonal to all the vectors in the set is the zero-vector.
- 7.
The norm structure is straightforwardly inherited from the norm that lies on the single-particle Hilbert space \(\mathcal {H}\).
- 8.
Note that the component \(\psi ^{(0)}\) on the zero-particle sector of \(\Psi \) is simply a complex number, describing the component of the wave function that does not contain any particle. Upon acting with the creation operator, a particle is created, hence there is no longer any fraction of the wave function that contains no particle, and the zero-particle component of \(\Psi \) acquires one particle with wave function \(\phi \).
- 9.
One can employ exactly the same argument for distinguishable, though identical, fermions.
- 10.
We will use the term modes to describe the degrees of freedom which are seen by each individual particle in the system. In other words, the single-particle Hilbert space \(\mathcal {H}\) can be seen as the structure that describes the quantum mechanics for these modes.
- 11.
Equation (7.43) can be thought of as a block diagonal representation. Therefore, we can restrict ourselves to considering \(a^{\dag }(f): \mathcal {H}^{(n)}\rightarrow \mathcal {H}^{(n+1)}\). We can now consider \(\Psi \in \mathcal {H}^{(n)}\) and \(\Phi \in \mathcal {H}^{(n+1)}\), with \(\Psi = \psi _1 \wedge \dots \wedge \psi _n\) and \(\Phi = \phi _1 \wedge \dots \wedge \phi _{n+1} \), and compute \(\left\langle a^{\dag }(f)\Psi ,\Phi \right\rangle = \sum _{\pi \in S_{n+1}} \left\langle f,\phi _{\pi (1)}\right\rangle \left\langle \psi _1,\phi _{\pi (2)}\right\rangle \dots \left\langle \psi _n,\phi _{\pi (n+1)}\right\rangle \). The adjoint is now defined by demanding that \(\left\langle a^{\dag }(f) \Psi ,\Phi \right\rangle = \left\langle \Psi ,a(f)\Phi \right\rangle \) for all \(\Psi \in \mathcal {H}^{(n)}\) and \(\Phi \in \mathcal {H}^{(n+1)}\). Because the Slater determinants form a total set, we can define the annihilation operator by means of its action on such vectors. The result (7.75) follows from the evaluation of the inner products.
- 12.
Thus, if an m-particle observable acts on an n-wave function, the result will be zero whenever \(n < m\).
- 13.
A word on notation: Whenever we consider an operator A, acting on some Hilbert space \(\mathcal {H}\), we denote \(A\mid _{\mathcal{K} \subset \mathcal {H}}\) the restriction of A on the subspace \(\mathcal{K} \subset \mathcal {H}\). Such a restriction simply limits the domain of the operator to the smaller subspace.
- 14.
Alternatively, see Dierckx et al. (2008) for applications in quantum information theory.
- 15.
In the next section, however, we sill see that Fock space is not as all-inclusive as it is often presented.
- 16.
In Sect. 2.3.1, we introduced the spectral theorem with the explicit demand that the spectral measure is projector-valued. This demand is, however, not strictly necessary and can be dropped to find a more general form of the spectral theorem. This constructions, in relation to quantum probability theory, are discussed in Holevo (2001) and can be connected to Naimark’s (\(\cong \) Neumark \(\mathrm {mod}\) transcription) dilation theorem (Neumark 1943; Stinespring 1955) which characterises positive-operator valued measures.
- 17.
Hence, we consider single-particle Hilbert space \(\mathcal {H}\cong \mathbb {C}\).
- 18.
Note that Mandel and Wolf (1995) uses different names for these states.
- 19.
In this case, we are just dealing with a parameter \(\zeta \in \mathbb {C}\), since we only consider a single-mode space.
- 20.
We already introduced this type of operator in Chap. 3 to describe time-reversal.
- 21.
- 22.
Also known as Q representation or Glauber Q representation.
- 23.
To extend this notion to a framework of many degrees of freedom, some effort is required to actually define this projector. The ultimate result (Fannes and Verbeure 1975; Petz 1990) for a single-particle Hilbert space \(\mathcal {H}\) of dimension d is given by
$$\begin{aligned} P_{\alpha } :=\frac{1}{\pi ^d}\int _{\mathcal {H}} \left\langle \Psi _{\alpha }, W(-\phi ) \Psi _{\alpha }\right\rangle W(\phi ) \mathrm{d}\phi . {(7.196)} \end{aligned}$$This result can also be extended to innfinite-dimensional single-particle spaces (Fannes and Verbeure 1975).
- 24.
- 25.
\(\mathbb {E}\) denotes the expectation value.
- 26.
The Q and W in equation (7.199) denote operators on Fock space and should not be confused with the Husimi function \(\tilde{Q}\) and Wigner function \(\tilde{W}\).
- 27.
In contrast to the coherent states of Sect. 7.6.2.
- 28.
See Appendix A for several formal definitions.
- 29.
A Wick monomial is a monomial of creation and annihilation operators, e.g. \(c^{*}(\psi _1)\dots c^*(\psi _n)c(\phi _m) \dots c(\phi _1)\) (Davies 1977).
- 30.
Truncated correlation functions are the multivariate version of cumulants and are also regularly referred to as “joint cumulants”.
- 31.
“\(c^{\#}\)” as a collective term to denote both “c” and “\(c^*\)”, e.g. each \(c^{\#}\) in (7.227) can be replaced by c or by \(c^*\).
- 32.
Technically, these are gauge transformations with respect to the group U(1), as commonly encountered in quantum electrodynamics (Cheng and Li 1984), where these gauge transformations refer to different choices of vector potentials. In principle, we can define more general gauge transformations with respect to other groups (Bratteli and Robinson 1987, 1997), but when we here refer to gauge-invariant states the relevant group is U(1).
- 33.
- 34.
- 35.
The consequence of this theorem is fundamental in the sense that it formalises the equivalence of Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics. Moreover, the fact that this theorem only holds for \(\dim \mathcal{S} < \infty \) implies that in quantum field theories this equivalence breaks down.
- 36.
Note that we introduced the quadratures (7.135) as induced by a specific representation. Since each state naturally gives rise to a GNS representation, we may label the quadratures by the state \(\omega \).
- 37.
Let us directly note that the term “Fock state” in mathematical physics does not refer to the “Fock state” in for example quantum optics. What quantum opticians refer to as a Fock state is rather a number state with respect to the Fock representation.
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Walschaers, M. (2018). Describing Many-Particle Quantum Systems. In: Statistical Benchmarks for Quantum Transport in Complex Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-93151-7_7
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DOI: https://doi.org/10.1007/978-3-319-93151-7_7
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93150-0
Online ISBN: 978-3-319-93151-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)