An intuitive construction of modular flow

The theory of modular flow has proved extremely useful for applying thermodynamic reasoning to out-of-equilibrium states in quantum field theory. However, the standard proofs of the fundamental theorems of modular flow use machinery from Fourier analysis on Banach spaces, and as such are not especially transparent to an audience of physicists. In this article, I present a construction of modular flow that differs from existing treatments. The main pedagogical contribution is that I start with thermal physics via the KMS condition, and derive the modular operator as the only operator that could generate a thermal time-evolution map, rather than starting with the modular operator as the fundamental object of the theory. The main technical contribution is a new proof of the fundamental theorem stating that modular flow is a symmetry. The new proof circumvents the delicate issues of Fourier analysis that appear in previous treatments, but is still mathematically rigorous.


Introduction
Given a Hamiltonian H, the density matrix ρ = e −βH tr(e −βH ) (1.1) is said to be thermal with respect to H at inverse temperature β.Conversely, given an invertible density matrix ρ, we can always construct a Hamiltonian with respect to which it is thermal, given by K ρ (β) = − 1 β log ρ.The operator is called the modular Hamiltonian of the state ρ.While K ρ is generally a highly nonlocal operator, it can be conceptually helpful to think of it as a physical Hamiltonian, since information-theoretic quantities associated to ρ can be treated as thermodynamic quantities associated to K ρ .
Many quantum systems of interest do not admit density matrices.Typical examples are thermodynamic systems at infinite volume [1][2][3] or local subregions in quantum field theory [4,5].Nevertheless, in these settings it is still possible to construct an operator that plays the same role played by K ρ in the finite-dimensional setting.The theory of this operator is known as Tomita-Takesaki theory or modular theory.It was first developed in [6], is treated in the textbooks [7][8][9], and has been reviewed for physicists in [10].In recent years it has reemerged in high energy physics as a tool for studying energy and entropy in quantum field theory and in semiclassical gravity; for an incomplete list of examples, see .
The setting for Tomita-Takesaki theory is a Hilbert space H with a von Neumann algebra A describing the quantum degrees of freedom in some subsystem.The traditional development of the theory begins with a state |Ω⟩ ∈ H that is "cyclic and separating" for A (defined in section 2.2), and defines an antilinear operator S Ω that acts as S Ω (a |Ω⟩) = a † |Ω⟩ , a ∈ A. (1.3) This is an unbounded operator, but it is sufficiently well behaved that the operator ∆ Ω ≡ S † Ω S Ω is densely defined, positive, and invertible.This is called the modular operator, and can be used to define the modular Hamiltonian K Ω ≡ − log ∆ Ω .The modular Hamiltonian generates a unitary group that can be used to act on operators in A via the map This is called called the modular flow of operators.The next step in the traditional development of the theory is to show that modular flow maps A into itself.This statement, known as Tomita's theorem, essentially says that the modular flow of a quantum subsystem is a symmetry of that subsystem.There are many approaches to proving Tomita's theorem [6,[39][40][41][42][43], but the general method is to take an integral transform of the function f (t) = ∆ −it Ω a∆ it Ω , show that the transformed function lies in A, and then to show that the inverse integral transform converges in a topology for which A is closed.After proving Tomita's theorem, one shows that the modular flow of operators satisfies something called the KMS condition, which basically means that the state |Ω⟩, restricted to the subsystem described by the algebra A, looks thermal with respect to the arrow of time generated by the Hamiltonian K Ω .
In this article, I present an approach to Tomita-Takesaki theory that differs from the one described in the preceding paragraphs.Pedagogically, the main difference is that I do not start with the antilinear operator S Ω , whose introduction I have always found fairly ad hoc.Instead, I start with the KMS condition, which characterizes thermality in terms of time-evolved two-point functions.I prove a uniqueness theorem, showing that if there exists some Hermitian operator H for which |Ω⟩ satisfies the KMS condition in subsystem A, then H must be the modular Hamiltonian K Ω . 1 In this way of developing the theory, thermal physics comes first: the modular Hamiltonian K Ω arises not as an abstraction, but as the only viable candidate for a thermal arrow of time.This gives a way of understanding why modular flow is so useful for studying quantum field theory and semiclassical gravity: if you want to apply thermodynamic reasoning to general states, then modular flow is the only tool available.
I then give a new proof of Tomita's theorem which differs from existing proofs, and which I hope is easier to understand.The proof works by showing that for any a in A and any b ′ in the commutant algebra A ′ , the commutator [∆ −it Ω a∆ it Ω , b ′ ] vanishes.By the bicommutant theorem of von Neumann, this implies that ∆ −it Ω a∆ it Ω must be in A. To show that the commutator vanishes, I construct a dense subset of operators a in A, called the "tidy subspace" of A, for which the map it → [∆ −it Ω a∆ it Ω , b ′ ] admits an analytic extension to the full complex plane, such that the norm of this analytic extension is bounded at infinity by an exponential function.I show that the analytic extension vanishes on the integers, and then apply a version of Carlson's theorem to show that the commutator is identically zero.This proves Tomita's theorem for the tidy subspace, and the general case is obtained by a continuity argument.
Since the goal of this paper is to make a complicated mathematical theory palatable and useful for theoretical physicists, I have included pedagogical treatments of background material in a review section and in several appendices.The outline of the paper is given in a bulleted list below.Minimally, I suggest reading section 3 and the preamble to section 4, and consulting section 2 on an as-needed basis to fill in mathematical background.This will give you the main ideas of the perspective I am taking on modular flow and on Tomita's theorem.The rest of the paper is more technical, and is aimed at readers who wish to work directly with modular flow themselves.
• In section 2, I give mathematical background that is needed for the rest of the paper.
I discuss several useful topologies on spaces of operators; I explain basic properties of the "cyclic and separating" states that are the continuum generalization of bipartite states with full Schmidt rank; I state some basic theorems concerning unbounded operators; and I explain the theory of contour integration and analytic continuation for operator-valued functions.
• In section 3, I introduce the KMS condition as a characterization of thermality in terms of two-point functions.I prove the KMS uniqueness theorem, introducing the modular operator by showing that for a cyclic and separating state, the modular Hamiltonian is the only Hamiltonian that could possibly make that state look thermal in the sense of KMS.I also observe that the modular Hamiltonian will automatically satisfy the KMS condition if Tomita's theorem holds.
• In section 4, I prove Tomita's theorem.I also prove the related statement that the modular conjugation operator maps A to A ′ .
• In three appendices, I explain Carlson's theorem, prove the basic properties of the Tomita operator, and outline the textbook proof of Tomita's theorem.
A shorter version of my proof of Tomita's theorem is presented in the companion paper [44], aimed at an audience of mathematicians.

Historical remarks
Four general proofs of Tomita's theorem have been given previously, by Takesaki [6], van Daele [39], Zsidó [40], and Woronowicz [43].Of these, van Daele's proof is the one that appears in textbooks (see e.g.[7,8]).There is also a proof of Tomita's theorem due to Longo [42] in the case where the von Neumann algebra A is hyperfinite.This property is believed to hold for the von Neumann algebras that appear in quantum field theory [45].
The proofs by Takesaki, van Daele, and Zsidó all make use of a lemma due to Takesaki concerning the resolvent of the modular operator.This lemma is also used in my proof, and is the subject of section 4.1.Immediately after introducing that lemma, van Daele's proof diverges from mine, and proceeds by using the resolvent lemma to study Fourier transforms of modular flow; the general outlines of this proof are explained in appendix C. Zsidó's proof, by contrast, uses Takesaki's resolvent lemma to construct a dense subset of A for which modular flow admits an analytic continuation to the full complex plane.I undertake a similar construction, but my dense subset is different from Zsidó's, and has the additional property that the analytic continuation of modular flow is bounded by an exponential function at infinity.This allows me to use Carlson's theorem to prove Tomita's theorem by studying the integer behavior of the analytic extension.By contrast, Zsidó's proof expresses analytic continuations of modular flow in terms of the theory of analytic generators [46], and appeals to results from that theory (essentially using Mellin transforms in place of van Daele's Fourier transforms) to finish the proof of Tomita's theorem.
The technique of applying Carlson's theorem to analytic extensions of modular flow was inspired by a proof of Tomita's theorem given by Bratteli and Robinson [47, pages 90-91] in the special case of a bounded modular operator.

Mathematical background
This is the most technical section of the paper, and unfortunately that is unavoidable, as one cannot understand the statement and proof of Tomita's theorem without first understanding the basics of operator theory.I have tried to write this section in a way that emphasizes the "rules of the game" for manipulating unbounded operators, without getting bogged down in specific technical lemmas.Eager readers may wish to proceed immediately to section 3 and consult this section on an as-needed basis.
Much of the material presented in this section can be found in the textbooks [8,48,49].Some of it is also discussed in my recent review article [50].

Operator topologies
Let H be a Hilbert space, and let B(H) denote the space of bounded operators on H.There are many interesting topologies on B(H).Among these, five appear most commonly: norm, ultrastrong, ultraweak, strong, and weak.The only topologies that appear in this paper are the norm, strong, and weak topologies.However, since the ultraweak and ultrastrong topologies are useful for studying operator algebras, and since several people have told me they find these topologies confusing, I have decided to include them in this section.Figure 1 norm ultrastrong ultraweak strong weak Figure 1: A flowchart of the standard topologies on operator algebras.Arrows point from stronger topologies to weaker topologies.So, for example, every sequence that converges in norm also converges ultrastrongly.
shows a flowchart of these topologies in order of strength.This figure should be understood to mean, for example, that any sequence convergent in the norm topology is convergent in the ultrastrong topology, but there may be sequences that converge in the ultrastrong topology which do not converge in the norm topology.
The norm topology is the easiest topology to understand.A bounded operator T : H → H has, by definition, some constant c satisfying The operator norm of T is defined as the infimum over all constants satisfying this inequality; equivalently, as The norm topology on B(H) is defined so that the net2 T α ∈ B(H) converges to T ∈ B(H) if and only if we have The norm topology is best thought of as a topology of uniform convergence: T α must converge to T uniformly on all vectors.The remaining four topologies are ultrastrong, ultraweak, strong, and weak.A useful mnemonic for understanding these topologies is as follows.
A "strong" topology is one where an operator is said to be small if its action on any state is small.A "weak" topology is one where an operator is said to be small if its expectation value in any state is small.The modifier "ultra" indicates that one must check smallness on mixed states as well as pure states; in a topology without the "ultra" modifier, smallness is determined only with respect to pure states.
A subalgebra A of B(H) that is closed under adjoints and that contains the identity is called a (unital) * -algebra.If A is closed in the norm topology, it is called a C* algebra.If it is closed in the weak topology, it is called a von Neumann algebra.From figure 1, we can see that the closure of a set in the norm topology is generally smaller than its closure in any other topology; so every von Neumann algebra is a C* algebra, but not every C* algebra is a von Neumann algebra.
Von Neumann algebras are the objects that naturally arise to describe the set of operators associated with a subregion of a physical system; see [10, section 2.6] for more discussion on this point.When studying von Neumann algebras, the most important theorem is von Neumann's double commutant theorem [51].The commutant of A, denoted A ′ , is defined as the set of all operators that commute with everything in A. The double commutant theorem is This is an extremely useful theorem, because to check that an operator is in A, it suffices to check that it commutes with every operator in A ′ .

Cyclic and separating states
As an example of a von Neumann algebra describing a quantum subsystem, we may think of the Hilbert space H = H 1 ⊗ H 2 , and take A to be the set of operators acting only on the first tensor factor: It is easy to check that this is a von Neumann algebra, and that its commutant is given by Given any vector |Ω⟩ ∈ H, one can produce a Schmidt decomposition: (2.7) The best-behaved states are those for which the Schmidt eigenvectors form complete bases of H 1 and of H 2 .These states are "fully entangled," and they have the property that the reduced density matrices on H 1 and H 2 are invertible.
For a general von Neumann algebra A, there is generally no associated tensor product structure, and one cannot talk about the Schmidt rank of a vector |Ω⟩ .There is, however, a more general criterion that captures the essential physics of a state having full Schmidt rank: this is the condition that |Ω⟩ is cyclic and separating.
Formally, the state |Ω⟩ is said to be cyclic for A if the set is dense in H.It is said to be separating for A if it is cyclic for A ′ , i.e., if the set It is a straightforward exercise to check that this property is satisfied by any full-Schmidt-rank state in a tensor product decomposition.A useful equivalent condition, which is not so hard to prove, is as follows.
Proposition 2.1.The state |Ω⟩ is separating for A if and only if it distinguishes operators on A in the sense that if an operator a ∈ A satisfies a |Ω⟩ = 0, then we must have a = 0.
The Reeh-Schlieder theorem [52][53][54] guarantees that in quantum field theory, states of physical interest are cyclic and separating.See [10, section 2] for a review of this point.

Unbounded operators
Tomita-Takesaki theory involves unbounded operators.This subsection provides a short exposition of the essential features of the theory of such operators.
Given a Hilbert space H and a Hilbert space K, an operator from H to K is defined to be a linear map T from a subspace D T ⊆ H, called the domain of T , into K.Note that this generalizes the standard definition of an operator by allowing T to act only on some subspace of H.This generalization is essential when considering unbounded operators, which may not be well defined for arbitrary input vectors.
An operator T is said to be bounded if it is continuous in the norm topology; i.e., if whenever |ψ n ⟩ ∈ D T converges to |ψ⟩ as a sequence of vectors, we have |ψ⟩ ∈ D T and T |ψ n ⟩ → T |ψ⟩ .For unbounded operators, even if the sequence |ψ n ⟩ converges, the sequence T |ψ n ⟩ need not converge.There is, however, a useful generalization of the boundedness condition which picks out a special class of unbounded operators.Definition 2.2.An operator T from H to K is said to be closed if, whenever |ψ n ⟩ ∈ D T converges to |ψ⟩ and T |ψ n ⟩ is a convergent sequence, then we have |ψ⟩ ∈ D T and T |ψ n ⟩ → T |ψ⟩ .An operator T from H to K is said to be preclosed or closable if it can be extended to a closed operator on a larger domain.
One can show (see for example [48, chapter 13]) that a preclosed operator has a unique smallest closed extension.If T is preclosed, then its smallest closed extension is denoted T .If the operator S 2 is an extension of the operator S 1 , we write S 1 ⊆ S 2 .
It is useful to keep in mind that if we have a closed operator T with domain D T , then for any subspace V ⊆ D T , the restriction T | V has a closed extension (namely, T ), and is therefore preclosed.If the subspace V is too small, then it might happen that the closure of T | V is a closed operator on some proper subspace of D T , so that we have T | V ⊊ T. By contrast, if V is a subspace such that T | V is equal to T , then V is said to be a core of T. Given a core for T , the action of T on any vector in its domain can be written as a limit of the action of T on vectors in the core.Consequently, to prove that a closed operator T satisfies some property, it is often sufficient to check that this property is satisfied on a core.This idea will be so important later on that we promote it to a formal definition for easy reference.Definition 2.3.Given a closed operator T with domain D T , a core for T is a subspace V ⊆ D T such that the restriction of T to V , which is a preclosed operator, has as its closure the full operator T .In an equation: (2.9) Remark 2.4.One useful way to think about closed operators is in terms of an object called a "graph."The graph of an operator T from H to K is a vector subspace of the Hilbert space H ⊕ K, and which is useful because all of the information about how T acts is contained in the structure of this subspace.It is defined by From what we have discussed so far, it is not so hard to see that T is closed as an operator if and only if Graph(T ) is topologically closed as a subset of H ⊕ K.The graph of the closure of an operator is equal to the topological closure of its graph.In particular, to show that a subspace V is a core for the operator T , it suffices to show that the set is dense in Graph(T ) with respect to the Hilbert space topology on H⊕K.If this perspective does not feel helpful, feel free to put it aside for now; we will use it exactly once in the paper, in one of the final steps of the proof of Tomita's theorem in section 4.2.Now we will discuss adjoints of unbounded operators.The operator T from H to K is said to be densely defined if the domain D T is dense in H.For any densely defined operator, we can define an adjoint operator T † from K to H.One must be a little careful in defining this, as T † will generally only be defined on some subspace of the Hilbert space K.We would like T † to satisfy the standard defining equation (2.12) To make sense of this equation, we should take the domain of T † to be the set of all vectors |ψ⟩ in K for which there exists a vector |ψ ′ ⟩ ∈ H satisfying There is a subtlety we must now address, which is that even if T is densely defined, its adjoint T † may not be densely defined, so we cannot always take the adjoint twice.This is addressed by the following proposition.(See for example [48, chapter 13].)Proposition 2.5 (Properties of adjoints).A densely defined operator T is preclosed if and only if its adjoint is densely defined.In this case, we have T † = (T ) † , and T † † = T .Furthermore, T † is always closed, and if S is an extension of T , i.e.T ⊆ S, then we have S † ⊆ T † .Now that we understand the properties of adjoints, we may define a Hermitian or selfadjoint operator as an operator from H to H satisfying T = T † .Note that for this equation to hold, the domains of T and T † must be the same!Note also that since adjoints are closed, any self-adjoint operator is automatically closed.For self-adjoint operators, even if they are unbounded, we have the following version of the spectral theorem.For a proof, see [48, chapters 12 and 13].
Definition 2.6.A subset of a topological space is Borel if it can be written using unions and intersections of at most countably many open or closed sets.
Definition 2.7.The spectrum of an operator T is the set of all complex numbers z for which z − T cannot be inverted as a bounded operator.Theorem 2.8 (Spectral theorem).Let T : D T → H be a self-adjoint operator in H. Then the spectrum of T is a subset of the real line, and to every Borel subset ω of the spectrum, there is an associated spectral projection P ω , which is a projection operator that commutes with T .Projectors for disjoint subsets of the spectrum project onto orthogonal subspaces, and the projector for a countable union of disjoint sets is formed by taking the countable sum of the projectors for each set.The spectral projection for the empty set projects onto the zero vector; the spectral projection for the full spectrum is the identity operator.
For any vectors |ψ⟩ , |ξ⟩ ∈ H, the function is a measure on the spectrum of T .From this, one may define an operator f (T ) for f any measurable, complex function of the spectrum.This operator is closed and densely defined, and has the following properties.
• The domain of f (T ) is the set of all vectors |ψ⟩ for which we have This integral gives the norm-squared of the vector f (T ) |ψ⟩ .
• The domain of f (T )g(T ) consists of vectors that are simultaneously in the domain of g(T ) and (f • g)(T ); on this domain we have f (T )g(T ) = (f • g)(T ).
• If a is a bounded operator that commutes with T on every vector where both aT and T a are defined, then a commutes with f (T ) on every vector where both af (T ) and f (T )a are defined.
The last thing we will need to know about unbounded operators is the existence of polar decompositions.To understand the statement of the theorem, it is helpful to know that a partial isometry from H to K is a map u : H → K for which u † u and uu † are both projections.For a proof of the following theorem, see e.g.[55, theorem 7.20].
Theorem 2.9.If T is a densely defined, closed operator from H to K, then T † T is positive and self-adjoint, and by the spectral theorem admits a positive square root |T | = √ T † T .The operator T can be written for some partial isometry u from H to K, and the projector u † u projects onto the orthocomplement of ker(|T |), which is the same as the orthocomplement of ker(T ).Moreover, the polar decomposition is unique: given any decomposition where v is a partial isometry, P is positive, and v † v projects onto the orthocomplement of the kernel of T , we must have v = u and P = |T |.
Definition 2.10.Given a von Neumann algebra A, we say a closed operator T is affiliated with A if it commutes with each a ′ ∈ A ′ on every vector where both a ′ T and T a ′ are defined.This is the closest we can get to an unbounded operator being "in" a von Neumann algebra.This last theorem is easy to prove using the bicommutant theorem and the uniqueness of the polar decomposition.It has the following useful corollary.
Corollary 2.12.If T is an unbounded operator affiliated with the von Neumann algebra A, then there exists a sequence T n of operators in A such that, for each |ψ⟩ in the domain of T , we have T n |ψ⟩ → T |ψ⟩ .
Proof.Write T = u|T |, and using the spectral theorem, take T n to be u|T | n , where |T | n is the projection of |T | onto the spectral range [0, n].

Analytic operator theory
To study Tomita-Takesaki theory, it is essential to understand when an operator-valued function of the complex plane can be thought of as holomorphic.The natural definition is the correct one -given a function f : C → B(H), we say it is holomorphic at z ∈ C if the limit exists.However, there is a subtlety: as explained in section 2.1, there are many important topologies on spaces of operators, and this limit may exist with respect to some of those topologies and not with respect to others.We therefore say, for example, that f is weakly holomorphic (or weakly analytic) at z if this limit exists in the weak topology, and norm holomorphic (or norm analytic) if this limit exists in the norm topology.Note that by figure 1, every norm analytic function is weakly analytic, but the converse is not guaranteed. 4any of the basic theorems of complex analysis hold for analytic operator-valued functions, regardless of which topology is used.The basic theorems of complex analysis all essentially follow from Cauchy's theorem that the integral of a holomorphic function around a simple closed curve vanishes.So to understand complex analysis for operator-valued functions, it is necessary to develop a theory of operator-valued integration.The theory of integration is easiest to understand for the norm topology on B(H), where the theory is known as Bochner integration.This theory is explained in the textbooks [56] and [57]; see also my recent blog post [58].The point is that for any Banach space X (e.g.B(H) or H itself), and for any measure space Ω, there exists a class of integrable functions f : Ω → X for which the integral Ω dµf can be defined.If Ω is a subset of some finite-dimensional real or complex space, then a continuous function is in the integrable class if and only if its norm has finite integral in the standard sense of integrals of real-valued functions.
The Bochner integral is linear, and it satisfies the triangle inequality The most important property of the Bochner integral is that if Y is a Banach space and ϕ : X → Y is a bounded linear functional of X, then we have In particular, this is true for bounded linear functionals ϕ : X → C. It is a basic fact of Banach space theory that a vector in Banach space is completely determined by the values it takes on bounded linear functionals.(I.e., if ϕ(x) vanishes for every bounded linear functional, then x is the zero vector.)Consequently, the integral dµ f is completely determined by the integrals dµ ϕ(f ), which are ordinary integrals of complex-valued functions.This lets one reduce the theory of contour integration in Banach space to the theory of contour integration for ordinary complex functions.For more details, consult the references mentioned above; for our purposes, it is enough to know that if an operator-or vectorvalued function of the complex plane is norm holomorphic, then its contour integral around any closed curve vanishes.From this one can prove the residue theorem, Morera's theorem (that a function is holomorphic in a domain if all closed contour integrals vanish), and the Weierstrass theorem (that a uniformly convergent sequence of holomorphic functions converges to a holomorphic function).Two essential property of operator-valued complex analysis are as follows.Detailed proofs can be found in [8, sections 2.28, 2.30] or [59].The statements are sketched in figure 2.
• If a bounded, positive operator T on H has spectrum bounded away from zero, then the function z → T z is norm analytic in the entire complex plane.This is easy to see, because it can be written in terms of an exponential as T = e z log T .
• If a bounded, positive operator T on H has spectrum going all the way to zero, then the function5 z → T z is norm analytic in the right half-plane and strongly continuous (but not norm continuous!) on the imaginary axis.This can be shown by projecting T onto some subset of its spectrum that is bounded away from zero, using the previous bullet point, and taking a limit while applying the theorem that uniformly converging limits of holomorphic functions are holomorphic.
Another important tool for studying bounded operators using complex analysis is the resolvent integral.If T is a bounded self-adjoint operator, then its spectrum is some bounded subset of the real line.Away from this subset, the function z → (z − T ) −1 is norm analytic.One can show that if f : C → C is a function that is analytic in a neighborhood of the spectrum of T , then the operator f (T ) -defined by the spectral theorem (theorem 2.8) -can be computed as a residue integral of the resolvent: In this equation, γ is a simple, closed, counterclockwise contour surrounding the spectrum of T , and contained in the domain of analyticity of f.The above considerations can be upgraded to unbounded operators.If T is a positive, self-adjoint, unbounded operator, then one can always define the operators T z using the If T is bounded and invertible, then the function T z is norm analytic in the entire complex plane.If it is bounded but not invertible, then T z is norm analytic in the right half-plane, and strongly continuous on the imaginary axis.spectral theorem, but these operators will mostly be unbounded, and their domains will all be different.It is therefore impossible to talk about the function z → T z being analytic in norm.What one can say instead is that for certain subsets of the complex plane, there exist vectors |ψ⟩ contained in the domain of T z for every z in this subset, and that within these subsets of the complex plane, the functions z → T z |ψ⟩ are analytic.
Let w = x + iy be a complex number, and suppose that the vector |ψ⟩ is in the domain of T w .Then, using the first bullet point of the spectral theorem (theorem 2.8), it is not hard to show that |ψ⟩ is in the domain of every T z where z lies in the vertical strip between the imaginary axis and w.See figure 3. The intuitive reason for this is that raising T to an imaginary power produces a bounded operator (in fact, a partial isometry), so changing the imaginary part of w does not affect the magnitude of the operator T w , and moving the real part of w closer to the imaginary axis makes T w less prone to diverge.One can show that the function z → T z |ψ⟩ is holomorphic in the interior of this strip, and continuous at the boundary. 6For a proof, see for example [8, section 9.15].Now, suppose that in addition to being positive, the unbounded operator T has trivial kernel.Even if the spectrum of T contains zero, so that T −1 is not invertible as a bounded operator, the spectral theorem (theorem 2.8) implies that when the kernel of T is trivial, the operator f (T ) is independent of the behavior of f at zero.So we may unambiguously define an unbounded operator T −1 by applying the function x → 1/x to T .The domain of T −1 is the closure of the image of T , and we have (2.24) When T is invertible, it is easy to show using the spectral theorem that T it is unitary, and so functions like z → T z |ψ⟩ are holomorphic in some strip and limit to unitary flows it → T it |ψ⟩ on the imaginary axis.It is possible to show from this the following beautiful The above discussion characterizes analytic extensions of unitary flows on vectors.It will also be important to understand analytic extensions of unitary flows on operators.
Theorem 2.14.Let T be a positive, invertible, unbounded operator.Let a be a bounded operator.If w is in the complex plane and the operator T w aT −w is densely defined and bounded on its domain, then the operator T z aT −z is densely defined and bounded on its domain for every z in the vertical strip between the imaginary axis and w.Consequently, each T z aT −z can be closed to a bounded operator.The function is defined on the strip, and is norm analytic in the interior and strongly continuous on the boundary.(See figure 5.) In fact, it is not necessary to check that T w aT −w is bounded on its full domain; it suffices to check that T w aT −w is defined and bounded on a core (definition 2.3) for the operator T −w .Conversely, if it → T it aT −it admits a norm-analytic continuation to the strip between the imaginary axis and w, such that this analytic continuation is strongly continuous on the ⟨ξ|T it |ψ⟩ ⟨ξ|T w |ψ⟩ Figure 4: A sketch of the statement of theorem 2.13.To check that a vector |ψ⟩ is in the domain of T w , it is sufficient to check that all functions of the form ⟨ξ|T it |ψ⟩ admit an analytic continuation from the imaginary axis to the vertical strip bounded by w.The overlap ⟨ξ|T w |ψ⟩ is obtained by evaluating this analytic function at w.The main difference from figure 3 is that it is easier to study the analyticity of complex-valued functions like ⟨ξ|T z |ψ⟩ than the analyticity of vector-valued functions like T z |ψ⟩ .Note that once we know |ψ⟩ is in the domain of T w , it follows from figure 3 that the vector-valued function also admits an analytic continuation.
T it aT −it T w aT −w Figure 5: A sketch of the statement of theorem 2.14.Given an unbounded, invertible, positive operator T , a bounded operator a, and a complex number w such that T w aT −w is densely defined and bounded on its domain, the function on the strip between the imaginary axis and w given by z → T z aT −z is norm analytic in the interior of the strip and strongly continuous on its boundary.On the imaginary axis, it takes the values T it aT −it .boundary of the strip, then T w aT −w must be densely defined and bounded on its domain, and the analytic continuation is given by z → T z aT −z .

Uniqueness of thermal symmetries
In a quantum system admitting density matrices, as explained in the introduction, the density matrix ρ is said to be thermal with respect to the Hamiltonian H at inverse temperature β if it has the form In order for this density matrix to make sense, the spectrum of H must be bounded below, and must be sufficiently tame so that e −βH has finite trace.The time-evolved two-point function of the thermal state can be written as Kubo, Martin, and Schwinger (KMS) [60,61] observed that this function admits an analytic continuation from it to more general complex z.Intuitively, we would like to simply make the substitution it → z, and write our analytic continuation as If H is bounded, then this is an analytic function of the entire complex plane.If H is only bounded below, however, then this function is not even well defined for arbitrary z.For Re(z) < 0 the operator e −zH diverges, and for Re(z) > β the operator e −(β−z)H diverges.However, in the vertical strip of the complex plane given by 0 ≤ Re(z) ≤ β, the function F (z) is well defined, and since it is written in terms of exponentials, it is analytic.The function F (z) therefore furnishes an analytic continuation of the two-point function to the strip of width β.Furthermore, the boundary values of this analytic continuation are given by and See figure 6.
The above observations motivate the following abstract definition of thermality for quantum systems described by von Neumann algebras.(i) H generates a symmetry of the state: for every real number t, we have (ii) H generates a symmetry of the algebra: for every real number t and every a ∈ A, we have e iHt ae −iHt ∈ A.
(iii) Two-point functions of A in the state |Ω⟩ look thermal with respect to the flow generated by H: for every a, b ∈ A, the function admits a bounded analytic continuation to the vertical strip 0 ≤ Re(z) ≤ β, and on the right boundary of this strip the analytic continuation is given by The point of this definition is that it has stripped away everything having to do with density matrices and tensor product decompositions.It lets us talk about thermal physics in systems without density matrices, such as quantum field theories, by expressing thermality as a property of the analytic structure of two-point functions.I find it helpful to think in terms of the following slogan.
Thermality on the lattice is always defined with respect to some Hamiltonian, and therefore with respect to some arrow of time.The KMS condition puts the arrow of time front and center: it requires that two-point functions evolved with respect to a given arrow of time have the same structure as the two-point functions of a Gibbs state in a lattice system evolved with respect to the lattice Hamiltonian.
Given a generic state |Ω⟩ and a generic von Neumann algebra A, two natural questions arise: • Does there exist a Hamiltonian for which the state |Ω⟩ looks thermal in the system A?
• Could there exist multiple different Hamiltonians with this property?
The first question is much harder to answer than the second.Using Tomita-Takesaki theory, we will ultimately see that the answer is yes, at least when the state |Ω⟩ is cyclic and separating (defined in section 2.2).But to motivate the introduction of Tomita-Takesaki theory, we will begin by answering the second question.We will show that if |Ω⟩ and A satisfy the KMS condition with respect to some Hamiltonian H, then that Hamiltonian must be the Tomita-Takesaki modular Hamiltonian K = − log ∆.This motivates the introduction of the modular operator directly from thermal physics, rather than introducing it as an abstract mathematical tool.
Consequently, H is given by − log ∆, where ∆ = S † S is the Tomita-Takesaki modular operator.
Proof.Fix two operators a, b ∈ A. By the KMS condition, there exists a function F (z), defined in the vertical strip 0 ≤ Re(z) ≤ 1, analytic in the interior of the strip and continuous on its boundary, and with boundary values See figure 7.
Intuitively, we would expect the function F (z) to be obtained by substituting it → z in equation (3.11); that is, we would like to have a formula like Unfortunately, this expression does not necessarily make sense; the Hamiltonian H is not assumed to be bounded below, so e −zH is an unbounded operator, and there is no guarantee that b |Ω⟩ is in the domain of e −zH .If we take this expression as a guiding principle, however, and evaluate it at z = 1, then by the KMS condition we expect there to be some sense in which the operator H satisfies ⟨Ω|ae −iHt b|Ω⟩ ⟨Ω|be iHt a|Ω⟩ 1 Again, this expression is not strictly correct, but it gives us an intuitive understanding of why the operator H is uniquely determined by the KMS condition -cyclicity of the vector |Ω⟩ implies that vectors of the form a |Ω⟩ are dense in H, so equation (3.14) provides a complete set of matrix elements for the operator e −H .Now, let us take our "morally correct" expression (3.14) and rewrite it as This expression is interesting because it is reminiscent of the identities satisfied by antilinear operators.An antilinear operator can be thought of as a map from H to the complex conjugate space H, which is a Hilbert space with the same vectors as H but with inner product ⟨ψ|ξ⟩ H = ⟨ξ|ψ⟩ H .
If e −H/2 were an antilinear operator mapping a † |Ω⟩ to a |Ω⟩ and b |Ω⟩ to b † |Ω⟩ , then we would have which is exactly equation (3.15).Now, while e −H/2 is not an antilinear operator, the above considerations suggest that it is worthwhile to define and study an antilinear operator that maps a † |Ω⟩ to a |Ω⟩ and b |Ω⟩ to b † |Ω⟩ .We define the antilinear operator S 0 on the domain D S 0 = A |Ω⟩ by Because |Ω⟩ is cyclic, the operator S 0 is densely defined, but generally unbounded.One can show (see appendix B) that because |Ω⟩ is cyclic and separating, the operator S 0 has the following properties.
• S 0 is preclosed.Its closure is denoted S, and is called the Tomita operator.
• Denoting the polar decomposition of S by the antilinear partial isometry J is called the modular conjugation and the operator ∆ is called the modular operator.The modular operator has trivial kernel, and is therefore invertible.The operator K = − log ∆ is called the modular Hamiltonian.
• The antilinear partial isometry J is in fact an antiunitary operator, and satisfies • The domain of S, which is the same as the domain of ∆ 1/2 , consists of all vectors of the form T |Ω⟩ such that either (i) T is in A, or (ii) T is affiliated to A and |Ω⟩ is in the domain of both T and T † .In either case, we have ST |Ω⟩ = T † |Ω⟩ .
Now, for every a, b ∈ A, the vectors a † |Ω⟩ and b |Ω⟩ are in the domain of ∆ 1/2 = e −K/2 .So the operator K satisfies This expression is exactly correct, and matches the "morally correct" expression (3.15) for the operator H.We will now use this observation to show ∆ = e −H , hence K = H.Note first that both ∆ and e −H are self-adjoint operators.It therefore suffices to show that e −H is an extension of ∆, since the inclusion and hence ∆ = e −H .So at a concrete level, the remaining proof reduces to showing that for every vector |ψ⟩ in the domain of ∆, the vector |ψ⟩ is also in the domain of e −H , and we have ∆ |ψ⟩ = e −H |ψ⟩ .
Let us consider an arbitrary vector in the domain of ∆.As explained in section 2.4, any vector in the domain of ∆ is also in the domain of ∆ 1/2 .As explained in the bulleted list above, any vector in the domain of ∆ 1/2 may be written as T |Ω⟩ , where T is affiliated to A and where |Ω⟩ is in the domain of both T and T † .We aim to show that T |Ω⟩ is in the domain of e −H .By theorem 2.13, this is equivalent to showing that for every |ξ⟩ ∈ H, the function it → ⟨ξ|e −iHt |T Ω⟩ (3.23) admits an analytic continuation to the strip 0 ≤ Re(z) ≤ 1.If |ξ⟩ is of the form a † |Ω⟩ for a ∈ A, then this analytic continuation exists by the KMS condition. 8The general case follows by taking limits using cyclicity of |Ω⟩ .So T |Ω⟩ is in the domain of e −H , and the action of e −H on that vector is given (via the KMS condition and theorem 2.13) by the formula ⟨a † Ω|e −H |T Ω⟩ = ⟨T † Ω|aΩ⟩. (3.24) But we also have So far, we have shown that if there exists a Hamiltonian H satisfying the KMS condition (definition 3.1), then it must be the modular Hamiltonian K.It is important to note that this does not tell us whether or not the modular Hamiltonian does satisfy the KMS condition; all we know so far is that the modular operator is the only operator that could work in principle.It is easy to see from the identity ∆ = S † S that we have ∆ |Ω⟩ = |Ω⟩ , so the modular Hamiltonian certainly satisfies property (i) of definition 3.1.The second property is the hard one to show -we must show that for a ∈ A, we have This is Tomita's theorem; the proof is quite involved, and is the subject of the next section.
To convince ourselves that it is actually worthwhile to prove this theorem, however, let us show that once we have proved Tomita's theorem, it will immediately follow that K satisfies condition (iii) of the KMS condition, and therefore does indeed provide a completely general, unique, thermal arrow of time for an out-of-equilibrium state.is bounded and analytic in the strip 0 ≤ Re(w) ≤ 1/2, and the function is bounded and analytic in the strip 0 ≤ z ≤ 1.
On the left side of the strip, we have So this function has the correct KMS boundary value on the left side of the strip.On the right side of the strip, we have Since we are assuming that Tomita's theorem holds, we have So we may use equation (3.20) to obtain So the function F (z) provides an analytic continuation of the two-point function satisfying the KMS condition.

Tomita's theorem, tidy operators, and the existence of thermal symmetries
Hopefully by now it is clear why we should care so much about proving Tomita's theorem.
In the previous section, we saw not only that the modular Hamiltonian is the unique Hamiltonian that could provide a general notion of thermal time, but also that Tomita's theorem is the only obstacle to guaranteeing that it does.This section presents a proof of Tomita's theorem.The idea of the proof is to study analytic continuations of maps like for a ∈ A. Since analytic functions are highly constrained, we will be able to use analyticity to show explicitly that all commutators of the form [∆ −it a∆ it , b ′ ] vanish when b ′ is an operator in the commutant.By the bicommutant theorem (cf.section 2.1), this will imply a ∈ A.
The structure of analytic continuations of maps like (4.1) is described by theorem 2.14.In particular, that theorem implies that the map in equation (4.1) admits an analytic extension to the entire complex plane if and only if, for every integer n, the operator ∆ −n a∆ n is densely defined and bounded on its domain.This is a big demand, and in fact we should not expect it to be true for arbitrary a ∈ A except in very special situations where the modular operator ∆ and its inverse ∆ −1 are bounded.So the way we will actually proceed is by constructing some dense subspace of A for which each ∆ −n a∆ n is bounded, and in fact for which the norm of ∆ −n a∆ n is bounded at infinity by an exponential function of n.I will call this the tidy subspace of A, denoted A tidy , and its members will be called tidy operators.I will prove Tomita's theorem for tidy operators, and obtain the general result by continuity.
The idea behind constructing tidy operators is that they should be operators for which the modular operator and its inverse "look bounded."Given 0 < λ 1 < λ 2 , the Heaviside theta function truncates the spectrum of ∆ to the range [λ 1 , λ 2 ].We will obtain tidy operators by starting with a vector like x |Ω⟩ for some arbitrary x ∈ A, then acting on this vector with Θ [λ 1 ,λ 2 ] (∆).
The resulting vector has no support in the spectral subspaces of ∆ near zero and infinity.We will show that this vector can be written as some other operator in A acting on |Ω⟩ via an equation like and take x [λ 1 ,λ 2 ] to be one of our tidy operators.Unfortunately, it takes some effort to get control over the operators Θ [λ 1 ,λ 2 ] (∆).It is much easier to control analytic functions of ∆, since these can sometimes be expressed as contour integrals using the resolvent of ∆.We will therefore proceed by developing a theory of analytic mollifiers, which are operators f (∆) for f an analytic function in a neighborhood of [0, ∞) such that f vanishes at infinity.We will obtain tidy operators by approximating step functions using analytic mollifiers.
It may be interesting to note that while Zsidó's proof [40] constructed analytic mollifiers like the ones I construct here, it was necessary in that case to appeal to the theory of analytic generators [46] to finish proving Tomita's theorem.This is because while one can use analytic mollifiers to produce operators a ∈ A for which ∆ −it a∆ it admits an entire analytic continuation, these analytic continuations are superexponentially growing at infinity, 9 and cannot be constrained using Carlson's theorem.
In section 4.1, I present a proof of a lemma due to Takesaki [6] concerning what happens when the resolvent of the modular operator is used as a mollifier.In section 4.2, I present a construction of the tidy subspace A tidy by extending arguments made in [40].In section 4.3, I show that for a a tidy operator and z ∈ C, the operator ∆ −z a∆ z is densely defined and bounded on its domain, and furthermore that when z is an integer, the closure of ∆ −z a∆ z lies in A. I then use Carlson's theorem to show ∆ −it a∆ it ∈ A for a in the tidy subspace, and obtain the statement for general a ∈ A by continuity.In section 4.4, I show that the techniques developed in this section also lead to an easy proof of the statement that for a ∈ A and J the modular conjugation, the operator JaJ is in A ′ .

Takesaki's resolvent lemma
The following lemma is extremely important in the development of the ensuing theory.Unfortunately, its proof is not particularly instructive -it involves a mathematical trick without any obvious physical meaning.Nevertheless, I hope that the reasons for trying to prove such a lemma are apparent from the preceding discussion, so the lack of insight provided by the proof will be acceptable, if undesirable.Lemma 4.1 (Takesaki's resolvent lemma).Let A be a von Neumann algebra, |Ω⟩ a cyclic and separating vector, and ∆ the associated modular operator.Fix x ′ ∈ A ′ .Then for every complex number z such that (z − ∆) is invertible as a bounded operator, we have for some unique x ∈ A. Furthermore, the norm of this operator satisfies the bound Proof.First note that because (z − ∆) −1 x ′ |Ω⟩ is in the domain of z − ∆, it is in the domain of ∆, and so in the domain of ∆ 1/2 .It follows (cf.appendix B) that it can always be written in the form x |Ω⟩ for some operator x affiliated to A. The whole work of the lemma is in showing that x is bounded.
To do this, it would suffice to show that x has bounded action on vectors of the form b ′ |Ω⟩ for b ′ ∈ A ′ .Unfortunately, showing this does not seem to be tractable in general.
Instead, we will write the polar decomposition of x as and try to show that x is bounded when acting on vectors of the form P |Ω⟩ where P is a spectral projection of |x|.In fact, because of the way the modular operator shows up in the lemma statement, it will be easier to show that x † is bounded when acting on vectors of the form Q |Ω⟩ for Q a spectral projection of |x † |.Once we have shown this, we will apply the fact that |Ω⟩ is separating to conclude that x is bounded, and derive the specific bound given in the statement of the lemma.
Let I be an interval in [0, ∞), and let Π I (|x † |) denote the spectral projection of |x † | in this range.Consider the vector Using the expression x † = |x|u † , and the easy-to-verify identity u|x|u † = |x † |, we can rewrite this vector as The norm-squared of this vector can be written as Using the fundamental identity (3.20) for the modular operator, we may rewrite this as By contrast, consider the action of the operator x ′ on the vector Π I (|x † |) |Ω⟩ .We have Taking the norm squared and expanding in terms of the inner product, we obtain (4.12)The last term in this expression can be compared to equation (4.10) to write (4.13) To make the first two terms look more like equation (4.10), we can use the universal inequality p 2 + |z| 2 q 2 ≥ 2|z|pq, which follows from the expression (p − |z|q) 2 ≥ 0. Using this gives Applying the Cauchy-Schwarz inequality then gives Finally, invoking equation (4.10) gives and hence Hence If the norm appearing in this inequality is nonzero, then we can divide by it on either side, and therefore obtain the inequality So if, by contrast, I is such that its left endpoint s 1 satisfies then we must have which, by the fact that |Ω⟩ is separating, implies and hence Π I (|x † |) = 0. We conclude that the spectral support of |x † | lies entirely within the interval lower bounded by zero and upper bounded by , and therefore that x † is bounded and we have as desired.

Constructing the tidy subspace
We will now use Takesaki's resolvent lemma, proved in the previous subsection, to study vectors like f (∆)a |Ω⟩ for certain functions f analytic in a neighborhood of [0, ∞).The idea will be to restrict our attention to functions f that die off sufficiently quickly at infinity, and then to show that for a ′ ∈ A ′ , we can write f (∆)a ′ |Ω⟩ as a contour integral where the integral has the properties of the Bochner integral on Hilbert space discussed in section 2.4.Using Takesaki's resolvent lemma (lemma 4.1), we will write (z for some operator a z ∈ A. This will let us express the contour integral as We will then show that this can be written in terms of an operator in A as a f |Ω⟩ .By judiciously choosing a sequence of analytic functions f k to approximate the Heaviside function, we will be able to construct an operator a [0,λ 2 ] satisfying the equation By invoking a symmetric argument, with the substitutions A ↔ A ′ and ∆ ↔ ∆ −1 , we will show that for any a ∈ A, there exists an operator a By combining equations (4.28) and (4.29), we will be able to show, for any a ∈ A, the existence of operators a We will use this to construct a dense set of states in H that have support in compact spectral ranges of ∆ and ∆ −1 , and that can be written either in terms of an operator in A acting on |Ω⟩ or in terms of an operator in A ′ acting on |Ω⟩ .10These vectors will even have the property that if we act on them with an integer power of the modular operator, ∆ n , they can still be written in terms of operators in A or A ′ , and that the norm of the resulting operator is bounded by some exponential function of n.
The operators in A that produce the special states described in the preceding paragraph will be called the tidy operators in A. The ones in A ′ will be called the tidy operators in A ′ .These can be thought of as the operators for which the modular operator "looks bounded." The rest of the section is written as a series of lemmas, propositions, and theorems.
Lemma 4.2.Let f be a bounded, analytic function in a neighborhood of [0, ∞), and let γ be a simple contour in that neighborhood surrounding [0, ∞) counterclockwise, and such that γ is contained in some bounded horizontal strip (so that the top and bottom parts of the contour do not get arbitrarily far away from each other at infinity).Suppose further that in the interior of the contour, f (z) vanishes uniformly in the limit Re(z) → ∞, and does so quickly enough that the real integral is finite, where s is the arclength parameter.Then for any vector |ψ⟩ , we have where this integral is evaluated in the sense of the Bochner integral from section 2.4.
Proof.Let Π n be the spectral projection of ∆ onto the range [0, n], and let ∆ n = ∆Π n be the restriction of ∆ to that spectral range.For each n, we can construct a vertical segment v n crossing the real line at n + 1/2 and cutting through the contour γ; see figure 8.We call the part of γ to the left of this vertical segment γ n .
It is a straightforward exercise using the spectral theorem to show the identity f (∆)Π n = f (∆ n )Π n .From this, and from the fact that each ∆ n is bounded, we may apply the residue formula (2.23) to obtain Since the spectrum of ∆ n is contained in the range [0, n], we may actually deform the contour to γ m + v m for any m ≥ n.It is easy to see from the assumptions of the lemma that the integral over v m vanishes in the limit m → ∞, so we may write The spectral theorem implies that in the limit n → ∞, the projections Π n converge strongly to the identity operator.So taking the n → ∞ limit on the left side of this equation gives the vector f (∆) |ψ⟩ .Taking the n → ∞ limit on the right side proves the theorem provided that we can interchange the limit and the integral.This interchange can be justified using a straightforward application of Lebesgue's dominated convergence theorem.
Proposition 4.3.Let f be a bounded analytic function in a neighborhood of [0, ∞) and γ a contour satisfying the conditions of the previous lemma.Suppose also that the function can be written as a f |Ω⟩ for some unique a f ∈ A. Furthermore, the norm of this operator is bounded by Proof.Since the function z 1/2 f (z) is bounded by assumption, the vector f (∆)a ′ |Ω⟩ is in the domain of ∆ 1/2 .As discussed in appendix B, this means there exists some operator Uniqueness is then an immediate consequence of the fact that |Ω⟩ is separating.Given b ′ ∈ A ′ , by the previous lemma and the properties of the Bochner integral discussed in section 2.4, we have By Takesaki's resolvent lemma (lemma 4.1), there exists some bounded operator a z ∈ A satisfying (z − ∆) −1 a ′ |Ω⟩ = a z |Ω⟩ .Using this identity gives The norm of the vector on the left-hand side satisfies the bound Applying the specific bound derived in lemma 4.1 gives the estimate The assumptions of lemma 4.2 guarantee that the integral is finite, so the action of a f on b ′ |Ω⟩ is bounded by a constant times ∥b ′ |Ω⟩∥.As explained in appendix B, the vectors b ′ |Ω⟩ form a core11 for affiliated operators constructed from the domain of ∆ 1/2 , so showing that a f is bounded on these vectors suffices to show that it is bounded on all vectors.
Theorem 4.4.Let Θ be the Heaviside theta function, with the convention Θ(0) = 1 2 .For λ 2 > 0 and a ′ ∈ A, there exists a unique a [0,λ 2 ] ∈ A satisfying For any nonnegative integer n, there also exists a unique a [0,λ 2 ],n ∈ A satisfying Furthermore, the norm of these operators is exponentially bounded in n, in that there exist λ 2 -independent constants α, β satisfying Proof.Note that because the step function Θ(λ 2 − ∆) cuts off the large spectral subspaces of ∆, each vector is in the domain of ∆ 1/2 , and hence (by the properties of the Tomita operator explained in appendix B) can be written uniquely as where a [0,λ 2 ],n is a potentially unbounded operator affiliated with A. We want to show that it is bounded and give an explicit bound on its norm as a function of n.This is an engineering problem.What we need is a sequence of functions f j , each analytic in a neighborhood of [0, ∞), and satisfying the conditions of proposition 4.3.We also want these functions to decay sufficiently quickly at infinity so that each for each nonnegative integer n, the functions z n f j (z) also satisfy the conditions of proposition 4.3.Finally, we want this sequence of functions to approximate the step function Θ(λ 2 − ∆) in the sense that we have This will allow us to write the following chain of equalities for any b ′ ∈ A ′ .
This lets us write the norm as So long as our functions f j are sufficiently well behaved that this limsup is bounded, we may conclude that a [0,λ 2 ],n is a bounded operator.But by the estimate on the operator norms given in proposition 4.3, it suffices to check lim sup The classic sequence of analytic functions approximating a step function is the sequence of sigmoid functions Indeed, it is a simple exercise using the spectral theorem to show that ∆ n f j (∆) converges strongly to ∆ n Θ(λ 2 − ∆). 12 So all we need to do is pick a good contour γ on which we can estimate inequality (4.49).A good contour is provided by combining the half-lines t + 2πi and t − 2πi for t ≥ 0 with the half-circle of radius 2π at the origin.See figure 9.
For the portion of the contour consisting of infinite half-lines, the function f j (z) is equal to 1 1+e j(t−λ 2 ) .For fixed n and large j, it converges to a step function in the range [0, λ 2 ].Consequently, the contribution of the half-lines to the contour integral in equation (4.49) gives 13   lim sup Under a fairly brutal approximation, we can bound this by Using another fairly brutal approximation, it is straightforward to see that the half-circle portion of the contour integral is bounded by

.54)
12 To see this concretely, fix |ψ⟩ ∈ H, write and apply the dominated convergence theorem to show that this goes to zero in the limit j → ∞. 13 I have been a little cavalier here about moving the limit lim sup j→∞ inside the integral, but this can be justified rigorously by a simple application of the dominated convergence theorem.
So the full contour integral is bounded by Since the norm a [0,λ 2 ],n is bounded by ∥a ′ ∥∞ 2π times this integral, it is easy to see that there exist some constants α, β satisfying Remark 4.5.By a completely symmetric argument to those given above, substituting ∆ ↔ ∆ −1 and A ↔ A ′ , it is easy to see that for λ 1 > 0, if we start with an operator a ∈ A, then there exists, for every integer n ≤ 0, an operator a and that these operators are exponentially bounded in |n|.
The set of all operators of the form a [λ 1 ,λ 2 ],n is called the tidy subspace A tidy .The vectors A tidy |Ω⟩ are dense in H.
Proof.If we have n ≥ 0, then we write If we have n < 0, then we write Applying remark 4.5 and then theorem 4.4 proves the first part of the theorem.Density of A tidy |Ω⟩ in A |Ω⟩ follows from the limit which is easy to show using the spectral theorem and invertibility of ∆.
Remark 4.7.The last theorem of this section tells us that A tidy |Ω⟩ is a core for any real power of the modular operator.(See definition 2.3 for the definition of a core.)This will be valuable in the next subsection for two reasons.First, we will use it to apply theorem 2.14, which lets us constrain operators of the form ∆ n a∆ −n in terms of their action on a core of ∆ −n .Second, we will use it to show the identity A ′′ tidy = A, which will allow us to obtain the general case of Tomita's theorem after proving that the theorem holds for tidy operators.
Proposition 4.8.For any real number x, the space A tidy Ω is a core for ∆ x .
Proof.We will use the characterization of a core given in remark 2.4.Namely, we will show that A tidy |Ω⟩ is a core for ∆ x by showing that vectors of the form a |Ω⟩ ⊕ ∆ x a |Ω⟩ , with a ∈ A tidy , are dense in the graph of ∆ x .
Suppose that |ψ⟩ is a vector in the domain of ∆ x such that ψ ⊕ ∆ x ψ is orthogonal to all such vectors.I.e., suppose that for all a ∈ A tidy , we have Our goal is to show that whenever this expression is satisfied, the vector |ψ⟩ vanishes.It suffices to show that the space (1 + ∆ 2x )A tidy |Ω⟩ is dense in Hilbert space.
To see this, note that per the construction of A tidy from theorem 4.6, each a ∈ A tidy satisfies an equation like aΩ = Θ(λ 2 − ∆)Θ(∆ − λ 1 )bΩ (4.65) for some 0 < λ 1 < λ 2 and some b ∈ A. The operator is bounded, injective, and invertible on the spectral subspace of ∆ corresponding to the range [λ 1 , λ 2 ].It therefore maps any dense subset of H into a dense subspace of that spectral subspace; in particular, the space is dense in the spectral subspace of ∆ for the range [λ 1 , λ 2 ] by the assumption that A |Ω⟩ is dense in H. Taking λ 1 → 0 and λ 2 → ∞ shows that (1 + ∆ 2x )A tidy |Ω⟩ is dense in the spectral subspace of ∆ for the range (0, ∞), which by invertibility of ∆ is equal to all of H.

Finishing the proof: analytic extensions of modular flow
Remark 4.9.We will now forget all of the details of the construction of the tidy subspace from the previous subsection, and use only the following facts.
• There is a subspace A tidy of A such that A tidy |Ω⟩ is dense in H, and is a core for each operator ∆ x .
• Each operator a ∈ A tidy has the property that a |Ω⟩ is supported in some spectral subspace of ∆ that is a closed interval in (0, ∞).
• Each vector a |Ω⟩ can be written as a ′ |Ω⟩ for some a ′ ∈ A ′ .
• Each vector ∆ n a |Ω⟩ for integer n can be written as a n |Ω⟩ = a ′ n |Ω⟩ for some a n ∈ A, a ′ n ∈ A. There exist constants α, β satisfying There is a piece of notation that was introduced in this list which we will use frequently in what follows, so we reemphasize it in the following definition.Definition 4.10.For an operator a in the tidy subspace, we will denote by a ′ the operator in A ′ satisfying a |Ω⟩ = a ′ |Ω⟩ .We will denote by a n and a ′ n the operators in A and Remark 4.11.We will now proceed to show that for a in the tidy subspace, the function it → ∆ −it a∆ it admits an entire analytic continuation that is exponentially bounded at infinity.We will need one lemma, which tells us how to think of the operators a † for a ∈ A tidy .We will then show the analytic continuations of modular flow exist, and reach the conclusion of Tomita's theorem by applying Carlson's theorem.
Lemma 4.12 (Dagger-Ladder Lemma).Let a be in the tidy subspace A tidy .Then we have Proof.Fix arbitrary b ∈ A, and write Since vectors of the form b|Ω⟩ are dense in H, this proves the lemma.
Theorem 4.13.If a is in the tidy subspace A tidy , then for integer values of n, the operator ∆ −n a∆ n is densely defined and bounded on its domain, with closure a −n .
Proof.First note that for any b in the tidy subspace, the vector b |Ω⟩ is in the domain of ∆ n .If we can show that a∆ n b |Ω⟩ is in the domain of ∆ −n , then we will have proved that b |Ω⟩ is in the domain of ∆ −n a∆ n .We will show that this is the case, and that we have Since proposition 4.8 tells us that A tidy |Ω⟩ is a core for ∆ n , we can then invoke theorem 2.14 to obtain the desired identity To proceed, note that we may write (4.74) so a∆ n b |Ω⟩ is in the domain of the adjoint of the Tomita operator S † , and we have Now applying lemma 4.12, we have So this vector is in the domain of S. Using the identity SS † = ∆ −1 , it follows that a∆ n b |Ω⟩ is in the domain of ∆ −1 , and we have Iterating this procedure n times, we see that a∆ n b |Ω⟩ is in the domain of ∆ −n , and that we have as desired.
Corollary 4.14.For a in the tidy subspace A tidy , the function is norm analytic in the entire complex plane.For integer values of z, the function F a is valued in A.
Proof.The preceding theorem tells us that at the integers, we have Analyticity of F a (z) follows almost immediately from theorem 2.14.Technically that theorem only guarantees that this function is analytic in the right half-plane and the left half-plane, and strongly continuous on the imaginary axis.But a simple argument using Morera's theorem shows that the function is analytic on the imaginary axis as well.(Anyway, this isn't so important, since we will eventually be applying Carlson's theorem, and Carlson's theorem works just fine for functions analytic in a half-plane.)Theorem 4.15 (Tomita's theorem on the tidy subspace).Let a be an operator in the tidy subspace A tidy Then for any t, the operator ∆ −it a∆ it is in A.
Proof.Fix a ∈ A tidy .It will suffice to show that for any b ′ ∈ A ′ , the operator ∆ −it a∆ it commutes with b ′ .Since a is in the tidy subspace, the previous corollary tells us that the function F a (z) = ∆ −z a∆ z (4.80) is norm analytic in the entire complex plane.For any b ′ ∈ A ′ , it follows that the function is norm analytic in the entire complex plane.Furthermore, it vanishes on the integers.The norm of this function is bounded by Since ∆ it is unitary for real t, we may bound this by Now, suppose O is an operator in A ′ tidy , a is an operator in A, and a n is a sequence of tidy operators converging as in the above equations.Fix b ′ , c ′ ∈ A ′ .We have

.89)
We already know via theorem 4.15 that Tomita's theorem holds for tidy operators.By applying this to b ′ , which is a tidy operator of A ′ , we observe that the operator (4.91)So ∆ −it a∆ it commutes with every tidy operator in A ′ .By proposition 4.17, it is in A.

A note on modular conjugations
In the preceding subsection, we showed that for a in the tidy subspace and for any z ∈ C, the operator ∆ −z a∆ z is closed and densely defined, and its closure is an operator in A.
Note that for a, b, c ∈ A, and S the Tomita operator, we have Since ∆ 1/2 a∆ −1/2 is bounded on its domain, this is a bounded function of b |Ω⟩ .It therefore follows that the operator SaS is bounded on its domain, so its closure is an operator in A ′ .
We have shown that for every operator a in the tidy subspace, the closure of the operator J(∆ 1/2 a∆ −1/2 )J is in A ′ .But since each operator b in the tidy subspace can be written as the closure of ∆ 1/2 a∆ −1/2 for some other operator a = ∆ −1/2 b∆ 1/2 in the tidy subspace, it follows that we have JbJ ∈ A ′ for each b in the tidy subspace.
For general a ∈ A, let b be an operator in the tidy subspace of A. We have since JbJ is in A ′ .So JaJ commutes with every tidy operator in A. By proposition 4.17, we conclude that JaJ is in A ′ .We have therefore shown the inclusion JAJ ⊆ A ′ .A symmetric argument gives JA ′ J ⊆ A, and we conclude JAJ = A ′ .
• The domain of S consists of all vectors of the form T |Ω⟩ such that either (i) T is in A, or (ii) T is affiliated to A with core A ′ |Ω⟩ , and |Ω⟩ is in the domain of both T and T † .In either case, we have ST |Ω⟩ = T † |Ω⟩ .An analogous statement holds for the domain of F.
• Denoting the polar decomposition of S by S = J∆ 1/2 , (B.4) the antilinear partial isometry J is called the modular conjugation and the operator ∆ is called the modular operator.The modular operator has trivial kernel, and is therefore invertible.Note that this equation implies the domain of ∆ 1/2 is the same as the domain of S.
• The antilinear partial isometry J is in fact an antiunitary operator, and satisfies J 2 = 1, hence J = J † .
• The polar decomposition of F is given by Note that it is often cleanest to think of an antilinear operator on H as an ordinary linear operator mapping a domain in H to the complex conjugate space H.We will not do this here, but if anything below seems confusing, the confusion can probably be resolved by thinking in these terms.As explained in section 2.3, an operator is preclosed if and only if its adjoint is densely defined.So we can show that S 0 is preclosed by finding a dense subspace of H on which its adjoint is defined.We will actually do this by showing S † 0 ⊇ F 0 , so that in particular S † 0 is defined on the domain of F 0 , which is A ′ |Ω⟩ , which is dense.A symmetric argument will imply F † 0 ⊇ S 0 , so we will also know that F 0 is preclosed.The properties of adjoints discussed in proposition 2.5 then guarantee S ⊆ F † , and we will eventually show equality.
To proceed, it will be helpful to recall the definition of the adjoint of an antilinear operator.If L is an antilinear operator with domain D L , then the adjoint L † is defined on every vector |ξ⟩ such that the map is bounded.The action of L † on any such vector is given via the matrix elements We will show that S † 0 can act on every vector of the form a ′ |Ω⟩ for a ′ ∈ A ′ , and that it acts on those vectors in the same way as F 0 .For any a ′ ∈ A ′ , and any a |Ω⟩ in the domain of S 0 , we have This is bounded as a function of a |Ω⟩ , so S † 0 can act on a ′ |Ω⟩ , and its action is determined by the matrix elements ⟨aΩ|S † 0 a ′ Ω⟩ = ⟨aΩ|(a ′ ) † Ω⟩. (B.9) This uniquely fixes S † 0 to act (antilinearly) on a ′ |Ω⟩ as S † 0 (a ′ |Ω⟩) = (a ′ ) † |Ω⟩ , which implies S † 0 ⊇ F 0 .
We now know that S 0 and F 0 are preclosed.We denote their closures by S and F .Since S † 0 is closed (cf.section 2.3), we have F ⊆ S † , and by proposition 2.5, we also have S ⊆ F † .To show equality, we will need to study the domain of S. This leads us to the following theorem.Note also that because T is affiliated to A, it is defined on A ′ |Ω⟩ , and can be uniquely determined if one requires that A ′ |Ω⟩ is a core for T .Conversely, if T is a closed operator affiliated to A with |Ω⟩ in the domain of both T and T † , then T |Ω⟩ is in the domain of S and satisfies ST |Ω⟩ = T † |Ω⟩.
Proof.We proceed by defining operators that act on A ′ |Ω⟩ the way we think T and T † should act, supposing they exist.Since T and T † are supposed to be affiliated with A, they should act on vectors of the form a ′ |Ω⟩ as Since we know F † ⊇ S, and |ψ⟩ is in the domain of S, this implies This gives us the inclusion α † 0 ⊇ β 0 , and a symmetric argument gives β † 0 ⊇ α 0 .From this we conclude that α 0 is preclosed, and we denote by T its closure.To see that T is affiliated with A, note that we have So T commutes with each a ′ ∈ A ′ on its core, and therefore commutes with each a ′ ∈ A ′ whenever the products T a ′ and a ′ T are both defined.The vector |Ω⟩ is in A ′ |Ω⟩ , so it is in the domain of both α 0 and β 0 , and therefore in the domain of both T and T † .We have Now we wish to use this theorem to upgrade the inclusion S ⊆ F † to an equality.Suppose a vector |ψ⟩ is in the domain of F † .We want to show that |ψ⟩ is in the domain of S, which means we want to show there exists an affiliated operator T like the one in the preceding theorem, with |ψ⟩ = T |Ω⟩ .It is straightforward to verify that in the proof of the preceding theorem, the operator T could have been constructed only under the assumption |ψ⟩ ∈ D F † , without the seemingly more stringent assumption |ψ⟩ ∈ D S .But the "converse" part of the above theorem then guarantees that we do in fact have |ψ⟩ ∈ D S .We therefore have S = F † , as desired.The above characterization of the domain of S also clearly implies It is a general fact -see section 2.3 -that any closed operator admits a unique polar decomposition.We write the polar decomposition of S as S = J∆ 1/2 , (B. 22) where ∆ 1/2 is a positive, self-adjoint operator given by ∆ 1/2 = √ S † S, and J is an antilinear partial isometry whose support is given by supp(J) = ker(S) ⊥ .The identity S 2 = 1 D S implies that the kernel of S is trivial, so J is supported on all of Hilbert space and is therefore an antiunitary operator.We also have that the kernel of ∆ is the same as the kernel of S, so ∆ is invertible on its domain.
We would now like to show the identities J 2 = 1 and J∆ 1/2 J = ∆ −1/2 .We first observe that thanks to the identity S 2 = 1 D S , we have It follows that every vector in the domain of J∆ 1/2 J is in the image of ∆ 1/2 , hence in the domain of ∆ −1/2 , so that the inclusion in expression (B.24) is an equality.To see J 2 = 1, we simply observe ∆ −1/2 = J∆ 1/2 J = J(JJ † )∆ 1/2 J = J 2 (J † ∆ 1/2 J). (B.27) The operator J † ∆ 1/2 J is positive, and the operator J 2 is unitary, so by uniqueness of the polar decomposition of the closed operator ∆ −1/2 , we must have J 2 = 1.

C van Daele's proof
This appendix gives a telegraphic outline of the structure of van Daele's proof of Tomita's theorem [39].This proof is very concise, but I find it unintuitive, which is why I presented a new proof in the main text of this paper.What follows is my best attempt to describe the techniques applied in van Daele's proof.The idea is that for a ∈ A, instead of showing that the modular flow ∆ −it a∆ it is in A, one considers the operator-valued function t → ∆ −it a∆ it , and tries to show that its Fourier transform p → dt e ipt ∆ −it a∆ it (C.1) is in A. One then tries to show that the Fourier inversion formula converges appropriately to guarantee ∆ −it a∆ it ∈ A.
One problem is the integral in equation (C.1) does not actually converge.But if we multiply ∆ −it a∆ it by a mollifying function f (t) that dies sufficiently quickly at infinity, then the Fourier transform p → dt e ipt f (t)∆ −it a∆ it (C.2) converges.If f (t) is nonzero, then the question of whether ∆ −it a∆ it is in A is exactly the same as the question of whether f (t)∆ −it a∆ it is in A, so the presence of the mollifying function does not affect the logic of the proof.A convenient choice of mollifying function is because this is periodic in the real direction of the complex plane, which can be used to evaluate the integral appearing in the Fourier transform by "closing a contour" and "picking up a pole."Concretely, the integral should in some sense be given by a residue at z = −1/2, which we expect to look like 2ie −p/2 ∆ −1/2 a∆ 1/2 .(C.5) These are integrals of unbounded operators, so one has to be careful about manipulating them, but by naively parametrizing the contour integral, one expects an expression of the following kind to hold: By being careful, one can show that the actual sense in which this expression is to hold is as an expression for matrix elements of vectors in an appropriate domain.In particular, for any vectors |ψ⟩ and |ξ⟩ that are in the domain of both ∆ 1/2 and ∆ −1/2 , one can show the identity e p/2 ⟨∆ 1/2 ψ|Q(p)|∆ −1/2 ξ⟩ + e −p/2 ⟨∆ −1/2 ψ|Q(p)|∆ 1/2 ξ⟩ = 2⟨ψ|a|ξ⟩. (C.9) From here, the basic idea of the proof is to show that this expression can be "solved" for Q(p) in that there is only one operator Q(p) satisfying this equation for any given p, and to explicitly show that Q(p) can be written as an operator in A obtained via a clever application of Takesaki's resolvent lemma (cf.section 4.1).
In a bit more detail, one actually considers the operator Q(p) = JQ(p)J, with J the modular conjugation, and studies the expression e p/2 ⟨∆ 1/2 ψ|J Q(p)J|∆ −1/2 ξ⟩ + e −p/2 ⟨∆ −1/2 ψ|J Q(p)J|∆ 1/2 ξ⟩ = 2⟨ψ|a|ξ⟩. (C.10) One then shows, using the expression S = J∆ 1/2 for the Tomita operator, that this equation is solved by an operator (a ′ ) † , where a ′ appears in a resolvent-lemma expression like for a p-independent real number c and a complex number λ that is some concrete function of p.To actually show this, one must approximate the vectors |ψ⟩ and |ξ⟩ by a sequence of vectors of the form (1 + ∆ −1 )b |Ω⟩ for b ∈ A, since these are the vectors for which van Daele's equation is under algebraic control.Once all this is done successfully (and once one also shows that van Daele's equation has a unique solution), it may be concluded that Q(p) is equal to (a ′ ) † .This tells us that the Fourier integral dt e ipt J∆ −it a∆ it J cosh πt (C.12) is in A ′ , and gives an expression satisfied by this operator in terms of the resolvent of ∆ −1 acting on the vector a |Ω⟩ .By being sufficiently careful about what it means to invert a Fourier transform of an operator-valued function, one can use this to show that J∆ −it a∆ it J is in A ′ for any t, so in particular for t = 0 we have JaJ ∈ A ′ .A symmetric argument shows JA ′ J = A, which gives ∆ −it a∆ it = J(J∆ −it a∆ it J)J ∈ JA ′ J = A. (C.13)

Theorem 2 . 11 .
If T is a bounded operator in the von Neumann algebra A, and its polar decomposition is T = u|T |, then we have u ∈ A, and for every bounded function f on the spectrum of |T |, we also have f (|T |) ∈ A. If T is a closed operator affiliated with the von Neumann algebra A, and the polar decomposition of T is u|T |, then we have u ∈ A, and for every bounded function f on the spectrum of |T |, we also have f (|T |) ∈ A.

Figure 3 :
Figure3: If T is an unbounded positive operator, and the vector |ψ⟩ is in the domain of T w , then it is also in the domain of T z for each z in the vertical strip between w and the imaginary axis.In the example sketched here, w has positive real part and the imaginary axis is the left boundary of the strip.The function z → T z |ψ⟩ is holomorphic in the interior of this strip and continuous on the boundary.

Definition 3 . 1 (Figure 6 :
Figure6: In a thermal state of inverse temperature β, the time-evolved two-point function can be analytically continued to a function in a vertical strip of width β.On the right edge of the strip, the function is equal to the "flipped" time-evolved two-point function.

Figure 7 :
Figure 7: Given a vector |Ω⟩ and an algebra A satisfying the KMS condition for the Hamiltonian H, the function ⟨Ω| ae −iHt b |Ω⟩ can be analytically continued to a vertical strip of width 1, with the analytic continuation being equal to ⟨Ω| be iHt a |Ω⟩ on the right edge of the strip.

. 25 )
Since vectors of the form a † |Ω⟩ are dense in Hilbert space, These two identities give the vector equation e −H |T Ω⟩ = ∆|T Ω⟩. (3.26)So every vector in the domain of ∆ is in the domain of e −H , and the actions of ∆ and e −H agree on such vectors.As explained above, this completes the proof of ∆ = e −H .

Theorem 3 . 3 (F ( 1 +
Modular KMS).Let H be a Hilbert space, A a von Neumann algebra, |Ω⟩ a cyclic and separating state, ∆ the modular operator, and K = − log ∆ the modular Hamiltonian.Suppose Tomita's theorem is true, so that we have e iKt ae −iKt ∈ A, a ∈ A.(3.28)Then K satisfies the third condition of the KMS condition.I.e., for every a, b ∈ A, the functionF (it) = ⟨Ω| ae −iKt b |Ω⟩ (3.29)admits a bounded analytic continuation to the vertical strip 0 ≤ Re(z) ≤ 1, and on the right boundary of this strip the analytic continuation is given by it) = ⟨Ω| be iKt a |Ω⟩ .(3.30) Proof.According to the properties of the modular operator described in appendix B, the states b |Ω⟩ and a † |Ω⟩ are in the domain of the operator ∆ 1/2 = e −K/2 .So, as explained in section 2.4 around figure 3, these states are in the domain of ∆ w for every 0 ≤ Re(w) ≤ 1/2, and the functions w → ∆ w b |Ω⟩ (3.31) and w → ∆ w a † |Ω⟩ (3.32) are bounded and analytic in this strip.Consequently, the function w → ⟨∆ w a † Ω|∆ w bΩ⟩ (3.33)

Figure 8 :
Figure8: Given a sufficiently nice contour γ enclosing the positive real axis [0, ∞], and given an integer n, one can draw a vertical segment v n passing through the real axis at n + 1/2, which splits the contour into a piece γ n to the left of the vertical segment, and a piece γ − γ n to the right of the vertical segment.

.48) γ 2π Figure 9 :
Figure 9: The specific contour used to evaluate the integral in equation (4.49), to show that operators in the tidy subspace satisfy a nice exponential bound.
.92) So SaS commutes with b when acting on c |Ω⟩ .Since vectors of the form c |Ω⟩ are dense in Hilbert space, we may conclude that SaS is an operator affiliated with A ′ .If a is in the tidy subspace, then we have SaSb |Ω⟩ = J∆ 1/2 a∆ −1/2 Jb |Ω⟩ .(4.93)
Theorem 3.2.Let A be a von Neumann algebra, and let |Ω⟩ be a cyclic and separating state for A. Suppose that H is a self-adjoint operator such that A, |Ω⟩, and H satisfy the KMS condition with β = 1.Then e −H/2 must be the positive part of the closed, antilinear operator S that acts on A |Ω⟩ as .83)The Phragmen-Lindelof theorem tells us that the norm of ∆ −Re(z) a∆Re(z)is upper bounded by the norm of a −n , where n is the nearest integer to Re(z) satisfying |n| ≥ |Re(z)|.We know by remark 4.9 that the norms of the operators a n are exponentially bounded.So F ab ′ is an entire function from C to B(H) that vanishes on the integers, and for which the norm is bounded by an exponential function of the nearest integer bounding the real part of z.A version of Carlson's theorem appropriate for this circumstance, given explicitly in appendix A, shows F ab ′ (z) = 0. Remark 4.16.Now we will show a sense in which A tidy generates all of A, and use this to prove Tomita's theorem in generality.Proof.The inclusion A tidy ⊆ A and the double commutant theorem A = A ′′ give the inclusion A ′′tidy ⊆ A. We must show the reverse inclusion.For this it suffices to show A ′ tidy ⊆ A ′ .To show this, recall that by proposition 4.8, A tidy |Ω⟩ is a core for ∆ 1/2 .It is therefore also a core for the Tomita operator S = J∆ 1/2 .By the properties of closed operators discussed in section 2.3, it follows that for any a ∈ A, there exists a sequence of tidy operators a n satisfying both Sa n |Ω⟩ → Sa |Ω⟩ = a † |Ω⟩ .(4.85)From this it is straightforward to show that for any operator b ′ in A ′ , we have the limits .88) Since A ′ |Ω⟩ is dense in H, this equation implies that [O, a] vanishes as an operator.So every element of A ′ tidy commutes with every element of A, as desired.Corollary 4.18 (The general version of Tomita's theorem).For any operator a ∈ A and any t ∈ R, the operator ∆ −it a∆ it is in A. Proof.Let b ′ be a tidy operator for the commutant algebra A ′ , and fix arbitrary |ψ⟩ , |ξ⟩ ∈ H.