Functional Analysis and Quantum Mechanics: an Introduction for Physicists Kedar S

Fortschritte der Physik Fortschr. Phys. 63, No. 9–10, 644–658 (2015) / DOI 10.1002/prop.201500023

Progress Paper Review of Physics Functional analysis and quantum mechanics: an introduction for physicists Kedar S. Ranade∗

Received 4 May 2015, revised 4 May 2015, accepted 8 June 2015 Published online 20 August 2015

It is the intention of this tutorial to introduce basic We give an introduction to certain topics from functional anal- concepts of functional analysis to physicists and to point ysis which are relevant for physics in general and in particular out their influence on quantum mechanics. We aim at for quantum mechanics. Starting from some examples, we dis- mathematical rigour in terminology, but we will leave out cuss the theory of Hilbert spaces, spectral theory of unbounded proofs. Our approach is not necessarily consistent with a operators, distributions and their applications and present mathematics textbook building up from bottom to top by proving every theorem from some basic axioms. Rather, some facts from operator algebras. We do not give proofs, but we strive at an intuitive understanding from concepts present examples and analogies from physics which should be and analogies known either from basic linear algebra or useful to get a feeling for the topics considered. used in basic quantum mechanics. Prerequisites for reading this tutorial are a knowledge of textbook quantum mechanics (such as wave mechanics and Dirac notation) including some basic quantum 1 Introduction optics (harmonic oscillator, ladder operators, coherent states etc.), but excluding quantum field theory; from It is well-known that physics and mathematics are mathematics, knowledge of analysis and linear algebra is closely interconnected sciences. New results in physics required.1 lead to new branches of mathematics, and new concepts from mathematics are used to describe physical situations and to predict new physical results. In the study 2 Hilbert space theory of physics quantum mechanics plays a fundamental role for almost all fields of modern physics: from quan- The formalism of quantum mechanics uses the concept tum optics, quantum information to solid-state physics, of a Hilbert space. In this section we will elaborate on the high-energy physics, particle physics etc. A similar role theory and give a full classification of Hilbert spaces, i. e. is enjoyed in mathematics by functional analysis in the we list (in some sense) all possible Hilbert spaces. The theory of differential equations, numerical methods and mathematics covered here can be found in several text- elsewhere. books on functional analysis, e. g. [1–3]. While every physicist knows quantum mechanics and every mathematician knows functional analysis, they often do not know the connection between these two fields—both of which were founded at the beginning of the 20th century. While every student of physics at- 1 This manuscript is partially based on a series of lectures enti- tends lectures on higher mathematics, such as analysis of tled Mathematische Aspekte der Quantenmechanik presented one or several variables, linear algebra, differential equa- (in german) at the Institut fur¨ Quantenphysik, Universität Ulm, tions and complex analysis, it is not so common that from April to August 2014. he gets to know functional analysis. Though it is possi- Institut fur¨ Quantenphysik, Universität Ulm, and Center for Inte- ble to understand the basics of quantum mechanics with grated Quantum Science and Technology (IQST), Albert-Einstein- pure linear algebra, a deeper understanding is gained Allee 11, D-89081 Ulm, Deutschland, Germany ∗ by knowing at least the very structure of functional Corresponding author: E-mail: [email protected], analysis. Phone: +49/731/50-22783

644 Wiley Online Library C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Fortschritte der Physik Fortschr. Phys. 63, No. 9–10 (2015)

Progress Paper Review of Physics

2.1 Introductory examples function on [0; ∞) with a possibly complex eigenvalue α, provided that Im α>0. How is this possible? We start this tutorial by presenting four examples, which show that the na¨ıve use of linear algebra fails in certain situations.2 2.1.4 Eigenvalues of hermitian operators II

In the previous example, one may argue that the radius 2.1.1 Commutators and traces cannot be negative and some part of the eigenfunction is cut off. Now we give a more dramatic example. First It is a simple exercise in linear algebra to show that we consider an abstract hermitian operator A with eigen- there holds Tr AB = Tr BA for any two matrices A and B vector v and eigenvalue λ: Av = λv. Hermiticity of A is (prove it by coordinate representation), which may be defined by Ax|y=x|Ay for all x, y ∈ H, and textbook rewritten as a trace of a commutator: Tr [A, B] = 0. In quantum mechanics (cf. e. g. Schwabl [10]) tells us that quantum mechanics, one requires, by analogy to the eigenvalues of hermitian operators are real by the follow- Poisson bracket of classical mechanics, [xˆ, pˆ] = i1H for ing simple argument: we calculate position and momentum operators xˆ and pˆ, respectively (known as canonical quantisation). Taking the trace on ∗ λ v|v=λv|v=Av|v=v|Av=v|λv=λv|v, (1) both sides of this equation yields 0 = i dim H.3 So does quantum mechanics really exist? so v|v = 0 yields λ = λ∗, and this implies λ ∈ R. Now take a spinless pointlike one-dimensional par- 2.1.2 Distributions ticle in position space, which is described by a square- integrable function on R. Consider the hermitian opera- ˆ = 3 + 3 = = ∂ Consider a particle in a box (square-well potential) with tor A xˆ pˆ pˆxˆ with the usual xˆ x and pˆ i ∂x and infinitely high walls: V (√x) = 0, if |x| ≤ a and V (x) =∞ take the function (plotted in the diagram) = 15 2 − 2 otherwise. Let (x) 4a5/2 (a x ) be the wavefunc- 1 −3/2 − 1 tion in the interior part which shall vanish outside f (x) = √ |x| e 4x2 . (2) 2 of the box. If we want to calculate the variance of the energy Hˆ =Hˆ 2−Hˆ2, we need the expectation Setting f (0) := 0, the function f is everywhere defined ˆ 2 ˆ 2 = 4 ∂4 = ˆ 2= R 4 value of H .FromH 4m2 ∂x4 0, we have H on and “smooth”. Further—unlike plane waves, a ∗(x)Hˆ 2(x)dx = 0. Since Hˆ2 is strictly positive, x=−a delta functions and the like—it is square-integrable 2 +∞ − − 1 the variance Hˆ is negative, which obviously is impossi- = 3 2x2 = and normalised: x∈R f (x) dx x=0 x e dx − 1 +∞ − 1 ble. What is wrong here? 2x2 = = 4x2 e x=0 1. For the derivative of g(x): e we find = − 1 −2 · = 1 −3 > g (x) ( 4 x ) g(x) 2 x g(x), thus for x 0, there 2.1.3 Eigenvalues of hermitian operators I holds −3 − / 3 − / − / x Consider a radially-symmetric potential V (r )inthree xˆ3pxˆ 3 2g(x) = x3 − x 5 2 + x 3 2 g(x) and i 2 2 space-dimensions. Using the ansatz (r ) = R(r)Y (ϑ, ϕ) lm (3) in spherical coordinates we get the spherical harmon- u(r) ics Y and by substituting R(r) = we get a radial − / d / lm r pˆxˆ3x 3 2g(x) = x3 2g(x) Schrodinger¨ equation for u(r) by adding a centrifugal i dx 2 + potential l(l 1) to V (r). We thus have a radial position −3 2mr 3 1/2 3/2 x = ∂ = x + x g(x). (4) operator rˆ and a corresponding momentum pˆr i ∂r , i 2 2 which fulfil the canonical commutator relation. Now, pˆr i α is hermitian, but ψ(r) = e r is a normalisable eigen- As f is even, and each power of xˆ and pˆ changes sym- ˆ = metries once, there holds Af i f for the function as a whole. Altogether, Aˆ is a hermitian operator with an 2 For these and other related examples see the articles by Gieres eigenfunction f ,butitseigenvalueisnotarealnumber. [4] and Bonneau et al. [5], which are very much recommended Where is the error? to the reader. 3 For a real commutator (without the imaginary prefactor i), 4 It can be arbitrarily often differentiated. It should not bother us one can use the ladder operators in a harmonic oscillator with that as a complex function there is an essential singularity in [aˆ, aˆ†] = 1 in a similar fashion. the origin, since we only consider real functions here.

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Progress Paper Review of Physics seen that it is not the different Hilbert spaces which dis- zero (otherwise divergent) and moreover, any series con- tinguishes different physical systems, but the operators structed out of these non-zero elements converges abso- 2 | | on these Hilbert spaces. lutely. The dimension of (I)thenis I , the scalar product x|y := x∗y . For example, the set of reals R is Theorem 1 (Classification of Hilbert spaces). For any pos- i∈I i i uncountable, so that 2(R) is inseparable. sible cardinality (i. e. “number of elements”) of a set there To conclude this section: In quantum mechanics we exists up to an isomorphism precisely one Hilbert space; in essentially deal with a single Hilbert space, the separa- particular, two Hilbert spaces are isomorphic, if and only ble, infinite-dimensional Hilbert space—though it is not if their dimensions coincide.7 the Hilbert space itself, but rather the structure of opera- The isomorphism concept can be understood as “in- tors on this space which distinguishes different physical distinguishable by inner structure”, i. e. by Hilbert space situations. operations only; in the well-known setting of square- integrable functions we may always think of all functions, some of which lie inside Hilbert space and others 3Spectraltheory outside (are “non-normalisable”), but in a pure Hilbert- space setting such considerations are not allowed. For a Spectral theory most prominently deals with the general- positive integer d,theHilbertspaceCd of column vec- isation of the spectral theorem from linear algebra—the tors with d complex entries is essentially the only Hilbert diagonalisation of hermitian (or normal) matrices—to space with dim H = d. A Hilbert space is called separa- infinite-dimensional spaces. To understand these things, ble, if its dimension is countable, i. e. less or equal of the a basic knowledge of measure and integration theory is cardinality of the integers.8 Thus, up to an isomorphism necessary, which will be introduced in the first two sub- there exists precisely one infinite-dimensional separable sections. Then we can finally state the precise meaning 2 R Hilbert space. of the Hilbert space L ( ). Consider for example a single particle in one space dimension: the Hilbert space is the set of square-integrable 3.1 Measure theory functions L2(R). Now change to the Fock basis, the eigenstates of the harmonic oscillator {|n| n ∈ N },whichis 0 Measure theory deals with the problem of associating a a countable and infinite basis. But obviously we have measure to certain sets.9 Measures can be thought of the by this an isomorphism to the Hilbert space 2(N )of 0 length of a distance, the area of a surface or the volume of square-integrable sequences. Note that this is still a ba- some space. Thus, given a set (think of Rn), one could sis regardless of the potential (though not necessarily an envision a function μ : P( ) → [0; ∞](whereP( )isthe eigenbasis), so that L2(R) is separable. Further, finite or power set, the set of all subsets of ) which maps to each countable unions and cartesian products of countable set its “volume”—which is non-negative. However, there sets are still countable; thus direct sums and tensor prod- are certain pitfalls, which are best illustrated by the fol- ucts of finitely and countably many separable Hilbert lowing paradox (in mathematics, this is related to the ax- spaces are separable. iom of choice). Although inseparable Hilbert spaces are not so common in physics, it is very easy to construct them. To con- Theorem 2 (Banach-Tarski paradox). Let A, B ∈ Rn be struct a Hilbert space for an arbitrary given dimension, arbitrary sets with non-empty interior. Then, there ex- , , ... choose an arbitrary index set I of that cardinality and let ist countably many sets A1 A2 and rigid mo- 10 , , ... = = tions m1 m2 ,suchthatA n∈N An and B 2 = ∈ CI | |2 < ∞ . (I): (xi)i∈I xi (5) ∈N mn(An), where both decompositions are disjoint, i. e. i∈I n Ai ∩ Aj =∅and mi(Ai) ∩ mj(Aj) =∅for i = j. For A, B ∈ The sum of possibly uncountably many terms is to be Rn bounded and n ≥ 3,thisisevenpossiblewithfinitely understood in the following sense of summability:it many sets and rigid motions. is defined only if at most countably many xi are non-

7 The proof is rather simple, it essentially reads Map one basis bi- 9 Measure theory is not to be confused with the theory of mea- jectively onto another. surements in quantum physics. 8 10 n This notion of separability is completely unrelated to the same These are bijective mappings mi on R which preserve term from entanglement theory. Euclidean distances, i. e. d(mi(x), mi(y)) = d(x, y).

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Progress Paper Review of Physics

Instead of always writing Lp( , , μ), one uses short- of Hilbert space, i. e. xˆψ/∈ H. At least such cases must hand notations like Lp( ). Note that the Hilbert space be excluded by restricting to an appropriate domain of L2( ) is different from the set of square-integrable func- definition D(A). tions L2( ). In particular, since a set with one element An operator A : D(A) → H with domain of definition {x} has Lebesgue measure zero, for a function f ∈ Lp( ), D(A) ⊆ H—which always shall be a dense subset of H— f (x) is strictly speaking not defined. Considering that is called hermitian (in mathematics the term symmetric in physics we only deal with integrals of L2 functions, is more common), if it is also resonable not to distinguish between functions which integrated over every subset give the same w|Av=Aw|v for all v, w ∈ D(A). (7) value. Due to the Hellinger-Toeplitz theorem, D(A) = H is in this case only possible, if A is bounded. The operators 3.3 Operators and their domains from quantum mechanics (xˆ, pˆ, Hˆ etc.) are mostly unbounded, and their domains are dense subsets of H. After introducing the Lebesgue integral, we can come The adjoint operator of Aˆ has (precisely) the domain of back to spectral theory. In the examples we presented a definition hermitian operator with an imaginary eigenvalue. On the other hand, one requires physical quantities like energy † D(A ):= {w ∈ H| v →w|Av is continuous on D(A)} , etc. to be real. The crucial point is that the terms “hermitian” and “self-adjoint” are different for unbounded (8) operators, while they coincide for matrices and, more † general, bounded operators. Only for the self-adjoint op- and is on this domain defined as follows: For w ∈ D(Aˆ ), erators the spectral theorem holds. the map v →w|Av can uniquely be extended to H and A linear mapping A : H → H on a Hilbert space can by the Riesz-Frechet´ theorem (see below), there exists a † either be bounded or unbounded. Such mapping is z ∈ H, such that z|v=w|Av;wethendefineA w := z. bounded, if the operator norm An operator is self-adjoint (rarely also hypermaximal hermitian), if A = A†, where in particular the domains of def- = † = Ax inition coincide, i. e. D(A) D(A ); only for these opera- A : sup (6) x =0 x tors the spectral theorem holds. The (algebraic) dual of a vector space E over some exists, i. e. it is finite. (For example, the operator norm of a field K is defined as the space of linear functionals E → hermitian matrix is the maximum of the absolute values K ; for a normed space the (topological) dual E of E con- of the eigenvalues.). sists of the functionals which are additionally required to It can be shown that linear operators are bounded, be continuous. if and only if they are continuous. In particular, all Theorem 4 (Riesz-Frechet´ theorem). Let H be a Hilbert linear operators on finite-dimensional Hilbert spaces— space and consider some ϕ ∈ H. The mapping ψ →ϕ|ψ essentially matrices—are bounded and thus continuous. is in H , and, on the other hand, every element of H can Examples of unbounded operators from physics are po- be written in this form by some ϕ ∈ H. sition and momentum operator and many Hamiltonians (free particle, harmonic oscillator etc.). In physics, one denotes elements of H as “ket” vec- Several theorems of linear algebra (in particular, the tors |ψ and elements of the dual H as “bra” vectors ϕ|; spectral theorem) can be extended to the set of bounded the stated theorem is the mathematical reason for Dirac’s operators (sometimes to a subset known as compact op- formalism. erators) and in the mathematical theory one only later discusses unbounded operators. We shall not present proofs here and thus deal only with the general case. 3.4 Multiplication operators We consider an unbounded linear operator A.Ingen- eral, such an operator is not defined on the Hilbert Given an operator A, it is not always easy to determine space, but only on a subset, its domain D(A). For ex- D(A†). An example, where the domain of definition of the ample, the position operator in positions representa- adjoint is straightforward to specify, is the case of mul- tion (xˆψ)(x) = xψ(x) may (by multiplication) throw some tiplication operators. Let ( , , μ) be a measure space, 2 ψ ∼ 1 | | = R L function with asymptotics (x) |x| for x 1out e. g. with the usual Lebesgue measure, and let

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Progress Paper Review of Physics representation) are said to be delta functions, because vice versa, for every T ∈ H there exists some ψ,suchthat ∂ δ = δ = = ψ xˆ (x) x (x), those of the momentum operator pˆ i ∂x T T . plane waves eikx. In both cases, these eigenfunctions are non-normalisable or not even true functions and thus To each element of a Hilbert space there is associated lie outside of the Hilbert space H = L2(R). Nevertheless, a unique element of its dual and vice versa, i. e. there is one regularly writes for position and momentum the some type of “symmetry” between H and H ;theyareiso- equalities H =∼ H morphic, , and “of the same size”. In case of the Hilbert space L2(R), the explicit form | | = | | = x x dx p p dp 1H (10) ϕ → ∗ϕ x∈R p∈R of the functional Tf is f (x) (x)dx.Now,ifwe choose a subset ⊆ H—limiting the elements ϕ allowed and in the domain of definition—, the number of admissible functionals increases. The space is in general not | | = , | | = , x x x dx xˆ p p p dp pˆ (11) a Hilbert space any more, and not all functionals can be x∈R p∈R written in the form of an integral. By an appropriately which do work well in calculations. Similarly, for a limited choice of (the test functions), we get a very potential well V (r ), one usually distinguishes between large set (the distributions, which “test” the test bound discrete and free continuous eigenstates and functions). A second aspect is that the functionals in often writes the identities need to be continuous; (sequential) continuity of f is the property that x → x0 in implies f (x) → f (x0) |EE| dE = 1H and E|EE| dE = Hˆ. (12) in C. If one would like to have many continuous functionals, there must be not too many convergent series But how to mathematically understand these things? In in . This will be achieved by an appropriate notion of δ the 1930s Dirac postulated a “delta function” with the continuity. δ = = δ = property that (x) 0forx 0, but (x)dx 1; for Before going into details, we shall give an overview on some other function f , this implies f (x)δ(x)dx = f (0). various mathematical structures in functional analysis in It is obvious that strictly speaking no such func- the following in Table 1; usually a set with some function tion δ can exist, but the concept is nevertheless has a specific name (such as Hilbert space). very useful in physics. The mathematical theory of For the details we must refer to the usual textbooks, such objects as the “delta function” is the theory of butwegivesomenoteshere: distributions (or generalised functions), which was devel- oped already in the 1940s by S. L. Sobolev and Laurent Schwartz. 1. A structure in an upper row induces a structure in a lower below; there sometimes is the possibility to go from a lower row to a higher one (parallelogram 4.1 Introduction equality, metrisability). 2. Scalar products and norms need a vector space in or- The basic idea of distributions in mathematics is, not der to be defined, metrics and topologies can be de- to consider the “delta function” and similar objects as fined on sets in general. functions but as functionals—objects mapping vectors 3. Weakening the axioms of a scalar product yields no- to scalars, as e. g. the delta function mapping the functions of bilinear and sesquilinear forms; without defi- tion f to a complex number f (0); in physics one would niteness norms and metrics become semi-norms and say that a delta function is defined only inside an inte- semi-metrics. (Note that the term metric is used in a gral. Mathematically, one can give a one-sentence defini- different fashion in differential geometry.) tion of distributions: they are the continuous linear func- 4. Completeness is a notion in metric spaces (because tionals on the space of test functions. Before explaining of Cauchy sequences). In topological spaces other no- these terms in detail, we shall rephrase the Riesz-Frechet´ tions of convergence (net or filter convergence, which theorem in a slightly different fashion. are equivalent) can be used. Theorem 5 (Riesz-Frechet´ theorem). Let H be a Hilbert 5. Characteristic notions for Hilbert spaces are orthogo- space and let H be its dual, i. e. the set of continuous lin- nality, for normed spaces duality (algebraic, topolog- ear functionals H → C. For every ψ ∈ H, the functional ical), for metric spaces sequences, Cauchy sequences

Tψ : H → C with Tψ (ϕ) =ψ|ϕ is an element of H ,and, etc.

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Progress Paper Review of Physics so for all ϕ ∈ D( ); the set D ( ) is then complete (in the 4.6 Tempered distributions and the Fourier transform sense of sequences). For test functions (or more generally for L1 and L2 functions) on Rn one can define the Fourier transform. To de- 4.4 Structure of distributions fine the Fourier transform for distributions, note that for L2(Rn) functions, there holds The set of distributions D ( ) obviously is a vector space. Moreover, every distribution can be multiplied with a (Ff )(x)g(x)dx = f (x)(Fg)(x)dx, (15) C∞-function f :(fT)(ϕ):= T(f ϕ), ϕ ∈ D( ), but two dis- 14 tributions cannot be multiplied. It is also possible to and one can, in the same spirit as in the case of deriva- define the convolution of two distributions, and distri- tives, try to define the Fourier transform by acting on the ∞ butions can be approximated by C -functions (e. g. the test function. However, Fϕ is never a test function, except delta distribution by Gaussians). for ϕ ≡ 0. Therefore, it is necessary to enlarge the space of Distributions can be differentiated arbitrarily: in test functions, which results in a smaller dual. The space 1 R ϕ = analogy to the product rule for a C ( ) functions, f which one considers is known as the space of rapidly de-

− f ϕ (the boundary terms vanish), on deﬁnes the creasing functions (or Schwartz functions): derivative of a distribution T by T (ϕ):=−T(ϕ ). Higher- ∞ order and higher-dimensional derivatives are accord- S(Rn):= f : Rn → C| f ∈ C (Rn), α |α| α ingly deﬁned by (D T)(ϕ):= (−1) T(D ϕ). α β (∀α, β ∈ N0) lim x D f (x) = 0 . (16) n→∞

4.5 Sobolev spaces These are the functions which are arbitrarily often differentiable and go to zero faster than every polynomial, ϕ ∈ Given a differentiable function f and any D( ), there and their derivatives should do the same. Here the stan- holds that f ϕ =− f ϕ (the boundary terms are zero). dard example is a Gaussian ϕ(x) = exp(−γ x), γ>0. The Even if f is non-differentiable, there may exist some important property with respect to Fourier transforms function g, such that f ϕ =− gϕ for all ϕ ∈ D( ). In is that F(S(Rn)) ⊆ S(Rn). There holds D(Rn) ⊆ S(Rn) and contrast to the common (strong) derivative, g is called the thus S (Rn) ⊆ D (Rn) – we shall not mention the topology weak derivative of f . The derivative in the sense of dis- here. The space S (Rn) is called the space of tempered distributions is still weaker. Altogether there are three no- tributions. tions of derivatives: the common (strong), the weak and The Fourier transform now is deﬁned by (FT)(ϕ):= the distributional derivative. T(Fϕ). There holds e. g. (FD(α)T)(ϕ) = (i/)|α|xαFT, Fδ = − i / π n/2 F = π n/2δ Rn α exp[ ax] (2 ) and 1 (2 ) 0 (in ), Example 1 (Derivatives). The absolute-value function where δ is the delta function at some point a ∈ Rn,i.e. | | a · : R → R is continuous for x = 0, but not differentiable. δ = δ − = n k a(x) (x a). For a polynomial f (x) k=1 akx there Nevertheless, there is a weak derivative, the sign function follows (Ff )(p) = n a ( ∂ )kδ. There is a regularisa- + > = − < k=1 k i ∂p sgn, which is 1 for x 0, 0 for x 0 and 1 for x 0. tion of distributions: for f ∈ S (Rn) there exists a function δ This function does not possess a weak derivative, but 2 is g and some α ∈ Nn, such that f = Dαg. its derivative as a distribution. 0

The space of m-times weakly differentiable Lp- functions (i. e. all Dαf exists for all α with |α| ≤ m), where 4.7 Applications all derivatives again lie in Lp, is called the Sobolev space W m,p( )(orHm,p( )), and p = 2 is often omitted. An Having discussed the guiding ideas in the deﬁnition interesting connection of weak and strong derivative is of distributions, we shall present some applications given by Sobolev’s lemma (or embedding theorem): Let of the theory of distributions, which are relevant to ⊆ Rn ∈ N > + n physics. be an open set, k 0 and m k p .Then, there holds W m,p( ) ⊆ Ck( ). 4.7.1 Differential operators

Rn → Consider a linear partial differential operator L : S( ) 14 Rn ∈ N → ∂ In physics, however, this is done in some instances. S( )oforderm given by T |α|≤m cα ∂α T with

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Progress Paper Review of Physics like block matrices. Commutative algebras are e. g. the Cauchy sequence), but weakly: for every bra-vector ψ|, set of diagonal matrices or continuous functions C[a; b] ψ|n converges to zero. on an interval or more generally on a compact set. An al- Given a sequence not of vectors but of operators, gebra which does not possess an identity is the continu- there are even more notions of convergence; we shall ous functions on an interval vanishing at the boundary consider only the three most important. Let (An)n∈N be a C0[a; b]. sequence in B(H), the bounded operators, and A ∈ B(H) An involution is a function † : A → A on an algebra A be an operator. fulfilling 1. The sequence (An)n∈N converges in the norm (or uni- †† 1. involutivity: x = x; formly)toA,iflimn→∞ An − A = 0inR.Here,A is 2. anti-linearity: (αx + βy)† = α∗x† + β∗y†; the as usual operator norm of eq. (6). 3. anti-multiplicativity (xy)† = y†x†. 2. It converges strongly (more precisely: with respect to the strong operator topology)tox, if for every x ∈ An involutive algebra is an algebra with an involution. H, the sequence (Ax ) ∈N in H converges strongly to In mathematics, usually a star is used instead of a dag- n n Ax ∈ H. ger (A∗ instead of A†), and such algebras are also called 3. It converges weakly (more precisely: with respect to ∗-algebra; this is the star in C∗-algebra. The standard ex- the weak operator topology)tox, if for every x ∈ H,the ample of involutive algebras are complex square matri- sequence (Ax ) ∈N in H converges weakly to Ax ∈ H. ces or bounded operators with their adjoints, i. e the n n transposed and complex conjugated matrix. The guid- Convergence in the norm implies strong, and this im- ing idea is to make terminology from spectral theory ab- plies weak convergence. To give counterexamples for the stract, i. e. to characterise them by their inner properties. opposite, consider An :=|nn| and Bn :=|n0|. Both se- The topic of operator algebras is connected to func- quences do not converge in the norm, but weakly to the tional analysis by normed algebras,i.e.algebraswitha null operator; the first one does so also strongly, the lat- norm. A Banach algebra (similarly to a Banach space) is ter does not. In the mathematical-physics literature there anormedalgebra,wherexy ≤ x y for all x, y ∈ A; often appear notions for which we shall give just the def- this condition implies continuity of the multiplication. initions, in order that the reader may grasp the idea, if he If there holds the (at first sight somewhat odd-looking) finds them somewhere (usually it is always good to think condition x†x = x2, the algebra is called C∗-algebra. of square matrices): (Using the operator norm this holds e. g. for square 1. A C∗-algebra is a a norm-closed involutive subalgebra matrices.) of B(H); 2. a von-Neumann algebra is a strongly (or equivalently weakly) closed involutive subalgebra of B(H). 5.2 Notions of convergence It can be shown that every C∗-algebra in this sense ful- In the preceding sections we already have mentioned the fils the C∗-property from above. The finite-dimensional possibility to use different notions of convergence. Here C∗-algebras—which always contain the identity—are we should consider in more detail some notions of con- easy to classify; they are the direct sums of simple vergence one usually encounters in mathematics litera- C∗-algebras, which again are the full matrix algebras ture. Let (xn)n∈N be a sequence and x be an element in Mn(C) of complex n × n-matrices. some Hilbert space H: The commutant of a set M ⊆ B(H)isthesubalge- bra M := {A ∈ B(H)| (∀B ∈ M)(AB = BA)} of all elements 1. The sequence (x ) ∈N converges strongly or in the n n in B(H) commuting with every element in M (the nota- norm to x,iflim →∞ x − x = 0inR. n n tion should not be confused with the dual); for exam- 2. It converges weakly to x, if for every y ∈ H the se- ple, as an instance of Schur’s lemma, if A = M (C), then quence y|x − x converges to zero in C. n n A = C1. The bicommutant theorem says that M is a von- In case of doubt, on usually takes strong conver- Neumann algebra, if and only if M = M . von-Neumann gence; more generally, weak convergence can be defined algebras can be decomposed into so-called factors (of on normed vector spaces and their dual. Strong conver- types In, I∞, II1, II∞ and III), where a factor is defined gence implies weak, but in infinite-dimensional spaces by M ∩ M = C1. not vice versa: consider an infinite-dimensional Hilbert Coming back to quantum mechanics: the postulates space, e. g. Fock space {|0, |1, |2, ...}, and a sequence of quantum mechanics say that there is a Hamiltonian xn =|n. This does not converge strongly (e. g. it is not a operator Hˆ, which generates the dynamics of a state

C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Wiley Online Library 655 Review Paper nmteaia em,tefunction the terms, mathematical In xs ntr operator unitary a exist | for continuity: Strong 2. ata isometries partial by root; bounded all square for operators same the the take do can and we theorem, it spectral the diagonalise can we so 0, operators an decomposition lar evolution H in tn’ hoe el sta hsi nedmath- indeed is this that us tells theorem Stone’s tion; generator mtclypsil.Teoeprmtrgopi not Hamiltonian oscil- harmonic is lator the group take norm-continuous: one-parameter necessarily The possible. ematically siltsabtaiyfs n hrfr sntcontinuous at g. not (e. is therefore and fast arbitrarily oscillates prtri laspstv eient.I h finite- the In semidefinite. case positive dimensional always is operator ntr-prtrvle ucino h tm)parameter (time) the of function unitary-operator-valued 656 the define can non- we of that set so the Schatten- discrete, g. is e. eigenvalues matrices, zero like behave in sense operators uniform compact some the The are operators. they finite-rank space of Hilbert im- limit the a of on closure compact; the is e., age i. compact, relatively is set pact called is operator An operators of Classes 5.4 on projections are holds there representation: polar into number complex A decomposition Polar 5.3 Schr the by vector Physics of Progress Physik der Fortschritte .Smgopproperty: Semigroup 1. e ob unitary. be to sen finite-dimensional, is space Hilbert z ˆ | n ∞ stm-needn,teslto steuiaytime unitary the is solution the time-independent, is ∈ t nteohrhn,gvnsc function a such given hand, other the On h oa eopsto o ttsta hr exist there that states now decomposition polar The = U and 0 ∈ R ( e t 0 + 0 R − t ) | and hc ufistefloigconditions: following the fulfils which ω = ( p | n B H + nrsa olw:tesnua ausof values singular the follows: as -norms ˆ 0). 1 ( / i ufie oso hsfor this show to suffices (it H ( 2) A t ϕ a ercvrdtruhfra deriva- formal through recovered be can t ) ). | = ∈ n ∈ B 0 2 [0; A n U ( e fa prtr The operator. an of H dne qaini equation odinger ¨ H | H samti,and matrix, a is − z .The ˆ ,sc that such ), i and sdfie as defined is ) Ht = π ˆ = compact .Ti sgnrlsdt the to generalised is This ). / a U U | V ω t + n ( ( ( → t t (0) + | b in n mapping ) + i 1 t B ∈ fteiaeo com- a of image the if , 0 s ofreeytime every for So . / n hr holds there , ) ( emi h exponent the in term 2 A C |+ H = = )(i.e. a edecomposed be may | U U A x 1 U | | / ( U and A ) where 2), t : t bouevalue absolute = | † ) ( d A | Ax d = U z t t | U ) √ | www.fp-journal.org = ( V 0). = (0) = s = † A )for = U r a echo- be can † | e e U A A − i ϕ and | to n this and , Ax ( i V H Ht t ˆ ˆ t for ) U U t .Ifthe , | | | | / Ax there A s ,the ( UU t → ( sa is ≥ po- r ) are ≥ t .If ) of = = 0 † . h sa omttr foeaos[ operators of commutators usual the easier thus handle. bounded, to are equation this in appearing operators the Further, states. coherent and oscillator monic oper- displacement the of case ators special a is this that Note hoe 7 Theorem unique essentially [13]. are operators these that says theorem e form: Weyl the in relations commutator get the (for- to can identity Baker-Campbell-Hausdorff we the hand, use other mally) the On uniquely. spectrum specify the not do relations commutator Heisenberg the that uhta o l i all for that such i lers u eae ocp,tegroup-theoretic of the concept, terms commutator related in a bracket but Lie algebras) as Lie known also mathematics (in B ru n ie dniy if identity, gives and group operators operator the hs prtr,frwihtenr sfiie omthe form finite, is norm classes Schatten the which for operators, Those tors erators ehnc.Ti hl epeetdhr nasotand short a in here presented be quantum shall of This formulation mechanics. mathematical a give to may used operators be of theory the from methods and Concepts momentum and position of Commutators 5.5 ipie a.I eea,oesat rmteHeisen- the [ from relations starts commutator one berg general, In way. simplified u [ but -omi nue yasaa product, scalar a by induced is 2-norm w ae fteeoeaosi n mode: one in operators these of cases two n H and h ievle of eigenvalues the p 1 r r A i i t .S aae ucinlaayi n unu ehnc—nintroduction mechanics—an quantum and analysis Functional Ranade: S. K. + x ˆ o i for i atcei o:Hletspace Hilbert box: a in particle a space Hilbert particle: free a h omttr sdi h elfr r not are form Weyl the in used commutators The h pcrmof spectrum The p e q 1 S i a s : D p ∞ = ˆ and ˆ ; j le’ inequality, older’s ¨ ( b S = ∈ = α

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Progress Paper Review of Physics relations eitAi eisAj = eisAj eitAi and eitBi eisBj = eisBj eitBi and the mathematical sense. However, not all mathematical states can be described this way, and the notion there is itA isB −istδ isB itA e i e j = e ij e j e i somewhat more general, but under some continuity assumptions, the converse also holds. 16 H are fulfilled. If the operators act irreducibly on , The states moreover form a convex and compact then there exists a unitary mapping U : H → L2(Rn), ϕ † set. Within these sets there exist extremal points, i. e. unique up to a phase ei ,suchthatUeitAi U = eitxî and † ∂ points which cannot be written as true convex combi- itBj = itpˆj = · = Ue U e where xî xi and pˆj ∂ are the usual i xi nations of other states. Formally: a state is extremal, if position and momentum operators on Rn. from ϕ = λϕ1 + μϕ2 with λ, μ > 0 and λ + μ = 1, it fol- The intuitive reason behind this is that the Heisen- lows that ϕ1 = ϕ2 = ϕ. Extremal points in the states space berg commutators exhibit the local properties (at every are the pure states, other states are mixed states. Accord- point x), but not the global ones. The Weyl commutators ing to the Krein-Milman theorem, a convex and com- take into account global properties; remember that the pact set (such as the states) is the closed convex span of exponentiated operators are translations of momentum its extremal states. For a two-dimensional Hilbert spaces and position, respectively, and thus connect different this is very well illustrated by the Bloch sphere. The in- points—in the case of the particle in a box, one would finitely many points on the surface correspond to the “hit the wall” if the shift distance is too large. (This is pure states, the convex hull (the interior) to the mixed related to the theory of Lie groups and Lie algebras.) states. (One can compare the example of a probability distribution with finitely many outcomes: there the extremal points are the probability distributions with 5.6 Mathematical structure of quantum mechanics one certain outcome.).

Thinking of the postulates of quantum mechanics (Hilbert space, state vector or density matrix etc.), one 5.7 More examples may ask what precise reason there is to use these postulates. There is a mathematical modelling of quantum It is beyond the scope of this introduction to present mechanics, which could explain this. Starting from the more than an overview of all the applications of modern assumption that quantum mechanics is a theory of what mathematics, i. e. still mathematics of the 20th century, is measurable, we must first state the measurable quanti- to physics. But at the end of this introduction we would ties (observables). A state of a system then is an object (a like to give some examples, where theorems from math- function) which maps to every observable its expectation ematics have direct application in physics: value. r The Schmidt decomposition was derived by Erhard The crucial assumption now is to describe observ- Schmidt in his 1906 doctoral thesis [14] and reap- ables by elements of an operator algebra with certain peared in Schrodinger’s¨ papers on entanglement in commutation relations. A state is a functional on the al- 1936 [15]. gebra of these operators, which should be continuous, r The idea of realising quantum operations as unitary linear, positive and normalised. A mathematical theo- operators on larger Hilbert spaces and Kraus opera- rem, the Gelfand, Neumark and Segal (GNS) represen- tors [16] is related to Stinespring’s dilation theorem tation, then tells us that, if these operators form a C∗- from 1955 [17]. Algebra and we have such a state, these operators may r The quantum de-Finetti theorem was derived by be represented by a subalgebra of the bounded linear Størmer in 1969 [18], then by Hudson and Moody [19] operators A = B(H) on some Hilbert space H. A state in and in physics by Caves, Fuchs and Schack [20]. mathematics is a functional ϕ : A → C which is positive r The proof of entropic uncertainty relations for Renyi´ (maps positive operators to non-negative numbers) and entropies by Maassen and Uffink [21] essentially re- which fulfils ϕ = ϕ(1) = 1 (the name of such a func- lies on the Riesz-Thorin interpolation theorem found tional even in mathematics is “state”). Comparing with in several textbooks. physics, a state is a density operator ρ; this operator generates a functional by ϕ(A):= Tr ρA, which is a state in 6 Summary 16 That is, if only {0} and H are invariant closed subspaces of all eitAi and eitBi . If this is not the case, the theorem can be It could not have been the aim of this introduction to give adapted accordingly. a complete overview of functional analysis or its use in

C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Wiley Online Library 657 Review Paper 658 References widely words. so Key not unfortunately is physicists. which by known mathematical but of general, amount physics and vast in mechanics a quantum is to relevant there knowledge that physics. clear for relevant is are It which analysis functional top- of some ics discussed we spirit this In physicists. for appropriate language a reader in the thinking mathematical give of to examples but mechanics, quantum and physics Physics of Progress Physik der Fortschritte 3 .B owy orei ucinlAnalysis Functional in Course A Conway, B. J. [3] mathe- modern of Methods Simon, B. and Reed M. [2] 5 .Bneu .Fru,adG Valent, G. and Faraut, J. Bonneau, G. [5] Gieres, F. [4] 7 .M efn n .G Kostyuchenko, G. A. and Gelfand M. I. [7] 1 .Wre,Fntoaaayi Srne 5 (Springer Funktionalanalysis Werner, D. [1] 6 .Esrd,Mß n nertoshoi Srne 4 (Springer Integrationstheorie und Maß- Elstrodt, J. [6] Srne 2 (Springer Analysis, Fourier II: Vol. (1975). Self-Adjointness 1980), (Revised edition Analysis enlarged Functional and I: Vol. physics, matical fPhysics of (2000). akSSSR Nauk 2005). ed., ucinlaayi,qatmmcais tutorial. mechanics, quantum analysis, Functional eot nPors nPhysics in Progress on Reports 69 nd 103 2–3 (2001). 322–331 , d,1990). ed., 4–5 15) rnltdi Izrail in translated (1955), 349–352 , mrcnJournal American www.fp-journal.org , 63 th ol Akad. Dokl. 1893–1931 , d,2005). ed., th 1]E Størmer, E. [18] (Springer, Stinespring, F. operations W. and effects, [17] States, Kraus, K. [16] Mathematicians for Schmidt, Theory E. Quantum [14] Hall, C. 2 B. (Springer 2 [13] Physik Theoretische der Scheck, F. Grundlagen Mathematische [12] Neumann, von J. [11] 2]C .Cvs .A uh,adR Schack, R. and Fuchs, A. C. Caves, M. Moody, C. R. [20] G. and Hudson L. R. [19] 1]E Schr E. [15] 2]H ase n .B .Ufﬁnk, M. B. J. and Maassen H. [21] 1]F cwb,Qatnehnk(pigr 6 (Springer, Quantenmechanik Schwabl, F. [10] 8 .M efn n .J iekn Verallgemeinerte Wilenkin, [9] J. N. G.G.Gould, and Gelfand M. I. [8] .S aae ucinlaayi n unu ehnc—nintroduction mechanics—an quantum and analysis Functional Ranade: S. K. (1955). 1983). (1936). 446–452 32, and (1935), 2013). (Springer, 2006). 1932). (Springer, Quantenmechanik 2002). 43 Gebiete verw. (1976). lichkeitstheorie 1310 (1988). 1103–1106 (1968). 1964). Deutscher Wissenschaften, (VEB der Verlag IV Vol. (Distributionen) Funktionen 1987), (Springer I Vol. 505–509. papers, Collected Gelfand, M. 5745 (2002). 4537–4559 , odinger, ¨ .Fnt Anal. Funct. J. ah Ann. Math. C 05WLYVHVra mH&C.Ka,Weinheim KGaA, Co. & GmbH Verlag WILEY-VCH 2015 rc abig hl Soc. Phil. Cambridge Proc. .Lno ah Soc. Math. London J. rc mr ah Soc. Math. Amer. Proc. 63 3 3–7 (1907). 433–476 , 86 (1969). 48–68 , hs e.Lett. Rev. Phys. .Wahrschein- Z. 33 .Mt.Phys. Math. J. 43 343–351 , 31 6 745–754 , 211–216 , 555–563 , th nd ed., ed., 60 ,