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September 8, 2017

Notes on von Neumann’s algebra formulation of Quantum Mechanics

Filed under: mathematics,science — ckrao @ 9:49 pm

The Hilbert space formulation of (non-relativistic) quantum mechanics is one of the great achievements of mathematical physics. Typically in undergraduate physics courses it is introduced as a set of postulates (e.g. the Dirac-von Neumann axioms) and hard to motivate without some knowledge of functional analysis or at least probability theory.  Some of that motivation and the connection with probability theory is summarised in the notes here – in fact it can be said that quantum mechanics is essentially non-commutative probability theory [2]. Furthermore having an algebraic point of view seems to provide a unified picture of classical and quantum mechanics.

The important difference between classical and quantum mechanics is that in the latter, the order in which measurements are taken sometimes matters. This is because obtaining the value of one measurement can disturb the system of interest to the extent that a consistently precise value of the other cannot be found. A famous example is position and momentum of a quantum particle – the Heisenberg uncertainty relation states that the product of their uncertainties (variances) in measurement is strictly greater than zero.

If measurements are treated as real-valued functions of the state space of system, we will not be able to capture the fact that the measurements do not commute. Since linear operators (e.g. matrices) do not commute in general, we use algebras of operators instead. We make use of the spectral theory leading from a special class of algebras with norm and adjoint known as von Neumann algebras which in turn are a special case of C*-algebras. The spectrum of an operator A is the set of numbers \lambda for which (A-\lambda I) does not have an inverse. Self-adjoint operators have a real spectrum and will represent the set of values that an observable (a physical variable that can be measured) can take. Hence we have this correspondence between self-adjoint operators and observables.

By the Gelfand-Naimark theorem C*-algebras can be represented as bounded operators on a Hilbert space {\cal H}. See Section II.6.4 of [3] for proof details. If the C*-algebra is commutative the representation is as continuous functions on a locally compact Hausdorff space that vanish at infinity. Furthermore we assume the C*-algebra and corresponding Hilbert space are separable, meaning the space contains a countable dense subset (analogous to how the subset of rationals are dense in the set of real numbers). This ensures that the Stone-von Neumann theorem holds which was used to show that the Heisenberg and Schrödinger pictures of quantum physics are equivalent [see pp7-8 here].

The link between C*-algebras and Hilbert spaces is made via the notion of a state which is a positive linear functional on the algebra of norm 1. A state evaluated on a self-adjoint operator outputs a real number that will represent the expected value of the observable corresponding to that operator. Note that it is impossible to have two different states that have the same expected values across over observables. A state \omega is called pure if it is an extreme point on the boundary of the (convex) space of states. In other words, we cannot write a pure state \omega as \omega = \lambda \omega_1 + (1-\lambda) \omega_2 where \omega_1 \neq \omega_2 are states and 0 < \lambda < 1). A state that is not pure is called mixed.

Now referring to a Hilbert space {\cal H}, for any mapping \Phi of bounded operators B({\cal H}) to expectation values such that

  1. \Phi(I) = 1 (it makes sense that the identity should have expectation value 1),
  2. self-adjoint operators are mapped to real numbers with positive operators (those with positive spectrum) mapped to positive numbers and
  3. \Phi is continuous with respect to the strong convergence in B({\cal H}) – i.e. if \lVert A_n \psi - A \psi \rVert \rightarrow 0 for all \psi \in H, then \Phi (A_n) \rightarrow \Phi (A),

then there is a is a unique self-adjoint non-negative trace-one operator \rho (known as a density matrix) such that \Phi (A) = \text{trace}(\rho A) for all A \in B(H) (see [1] Proposition 19.9). (The trace of an operator A is defined as \sum_k \langle e_k, Ae_k \rangle where \{e_k \} is an orthonormal basis in the separable Hilbert space – in the finite dimensional case it is the sum of the operator’s eigenvalues.) Hence states are represented by positive self-adjoint operators with trace 1. Such operators are compact and so have a countable orthonormal basis of eigenvectors.

When \rho corresponds to a projection operator onto a one-dimensional subspace it has the form \rho = vv^* where v \in {\cal H} and \lVert v \rVert = 1. In this case we can show \text{trace}(\rho A) = \langle v, Av \rangle = v^*Av, which recovers the alternative view that unit vectors of {\cal H} correspond to states (known as vector states) so that the expected value of an observable corresponding to the operator A is \langle v, Av \rangle. This is done by choosing the orthonormal basis \{e_k \} where e_1 = v and computing

\begin{aligned} \text{trace}(\rho A) &= \sum_k \langle e_k, vv^*Ae_k \rangle\\ &= \sum_k e_k^* v v^* Ae_k\\ &= e_1^* e_1 e_1^*Ae_1 \quad \text{ (as }e_k^*v = \langle e_k, v \rangle = 0\text{ for } k > 1\text{)}\\ &= e_1^*Ae_1\\ &= \langle v, Av \rangle. \end{aligned}

Trace-one operators \rho can be written as a convex combination of rank one projection operators: \rho = \sum \lambda_k v_k v_k^*. From this it can be shown that those density operators which cannot be written as a convex combination of other states (called pure states) are precisely those of the form \rho = vv^*. Hence vector states and pure states are equivalent notions. Mixed states can be interpreted as a probabilistic mixture (convex combination) of pure states.

Let us now look at the similarity with probability theory. A measure space is a triple (X, {\cal S}, \mu) where X is a set, {\cal S} is a collection of measurable subsets of X called a \sigma-algebra and \mu:{\cal S} \rightarrow \mathbb{R} \cup \infty is a \sigma-additive measure. If g is a non-negative integrable function with \int g \ d\mu = 1 it is called a density function and then we can define a probability measure p_g:{\cal S} \rightarrow [0,1] by

\displaystyle p_g(S) = \int_S  g\ d\mu \in [0,1], S \in {\cal S}.

A random variable f:X\rightarrow \mathbb{R} maps elements of a set to real numbers in such a way that f^{-1}(B) \in {\cal S} for any Borel subset of \mathbb{R}. This enables us to compute their expectation with respect to the density function g as

\displaystyle \int_X f \ dp_g = \int_X fg\ d\mu.

This is like the quantum formula \text{Tr}(\rho A) with our density operator \rho playing the role of g and operator A playing the role of random variable f. Hence a probability density function is the commutative probability analogue of a quantum state (density operator).

While Borel sets are the events from which we define simple functions and then random variables, in the non-commutative case we define operators in terms of projections (equivalently closed subspaces) of a Hilbert space {\cal H}. A projection operator P is self-adjoint, satisfies P^2 = P and has the discrete spectrum \{0,1\}. Hence they are analogous to 0-1 indicator random variables, the answers to yes/no events. For any unit vector v \in {\cal H} the expected value

\displaystyle \langle v, Pv \rangle = \langle v, P^2v \rangle = \langle Pv, Pv \rangle = \lVert Pv \rVert^2

is interpreted as the probability the observable corresponding to P will have value 1 when measured in the state corresponding to v. In particular this probability will be 1 if and only if v is in the invariant subspace of P. We define meet and join operations \vee, \wedge on these closed subspaces to create a Hilbert lattice ({\cal P}({\cal H}), \vee, \wedge, \perp):

  • A \wedge B = A \cap B
  • A \vee B = \text{closure of } A + B
  • A^{\perp} = \{u: \langle u,v \rangle = 0\ \forall v \in A\}

Borel sets form a \sigma-algebra in which the distributive law A \cap (B \cup C) = (A \cap B) \cup (A \cap C) holds for any elements of {\cal S}. However in the Hilbert lattice the corresponding rule A \wedge (B \vee C) = (A \wedge B) \vee (A \wedge C) (where A, B, C are projection operators) only holds some of the time (see here for an example). This failure of the distributive law is equivalent to the general non-commutativity of projections.

A quantum probability measure \phi:{\cal P} \rightarrow [0,1] can be defined by combining projections in a \sigma-additive way, namely \phi(0) = 0, \phi(I) = 1 and \phi(\vee_i P_i) = \sum_i \phi(P_i) where P_i are mutually orthogonal projections (P_i \leq P_j^{\perp}, i \neq j). Gleason’s theorem says that for Hilbert space dimension at least 3 a state is uniquely determined by the values it takes on the orthogonal projections – a quantum probability measure can be extended from projections to bounded operators to obtain \phi(A) = \text{Tr}(\rho_{\phi} A), similar to how characteristic functions are extended to integrable functions. Hence this is a key result for non-commutative integration (note: the continuity conditions defining \Phi in 1-3 above are stronger). We choose von Neumann algebras over C*-algebras since the former contain all spectral projections of their self-adjoint elements while the latter may not [ref].

So far we have seen that expected values of observables A are derived via the formula \text{Tr}(\rho A). To derive the distribution itself, we make of the spectral theorem and for self-adjoint operators with continuous spectrum this requires projection valued measures. A self-adjoint operator A has a corresponding function E_A:{\cal S} \rightarrow {\cal P}({\cal H}) mapping Borel sets to projections so that E_A(S) represents the event that the outcome of measuring observable A is in the set S: we require that E_A(X) = I and S \mapsto \langle u,E_A(S)v \rangle is a complex additive function (measure) for all u, v \in {\cal H}. We use E_A(\lambda) as shorthand for E_A(\{x:x\leq \lambda\}). Similar to the way a finite dimensional self-adjoint matrix M may be eigen-decomposed in terms of its eigenvalues \lambda_i and normalised eigenvectors u_i as

\begin{aligned} M &= \sum_i \lambda_i u_i u_i^T \\ &= \sum_i \lambda_i P_i \quad \text{(where }P_i := u_i u_i^T \text{ is a projection)}\\ &= \sum_i \lambda_i (E_i - E_{i-1}), \quad \text{(where } E_i := \sum{k \leq i} P_k\text{ ),} \end{aligned}

the spectral theorem for more general self-adjoint operators allows us to write

A = \int_{\sigma(A)} \lambda dE_A(\lambda)

which means that for every u, v \in {\cal H},

\langle u, Av \rangle = \int_{\sigma(A)} \lambda d\langle u,E_A v \rangle.

Here, the integrals are over the spectrum of A. Through this formula we can work with functions of operators and in particular the distribution of the random variable X corresponding to operator A in state \rho will be

\text{Pr}(X \leq x) = E\left[ 1_{\{X \leq x\} }\right] = \text{Tr} \left( \rho\int_{-\infty}^x dE_A(\lambda) \right) = \text{Tr} \left( \rho E_A(x) \right).

The similarities we have seen here between classical probability and quantum mechanics are summarised in the table below, largely taken from [2] which greatly aided my understanding. Note how the pairing between trace class and bounded operators is analogous to the duality of L^1 and L^{\infty} functions.

Classical Probability
Quantum Mechanics
(non-commutative probability)
(X,{\cal S}, \mu) – measure space ({\cal H}, {\cal P}({\cal H}), \text{Tr}) – Hilbert space model of QM
X – set {\cal H} – Hilbert space
{\cal S} – Boolean algebra of Borel subsets of X called events {\cal P}({\cal H})orthomodular lattice of projections (equivalently closed subspaces) of {\cal H}
disjoint events orthogonal projections
\mu:{\cal S} \rightarrow {\mathbb R}^{+} \cup \infty\sigma-additive positive measure \text{Tr} – functional
g \in L^1(X,\mu), g \geq 0, \int g \ d\mu = 1 – integrable functions (probability density functions) \rho \in {\cal T}({\cal H}), \rho \geq 0, \text{Tr}(\rho) = 1 – trace class operators (density operators)
p_g(S) = \int \chi_S g\ d\mu \in [0,1], S \in {\cal S}probability measure mapping Borel sets to numbers in [0,1] in a sigma-additive way \phi(S) = \text{Tr}(\rho_{\phi } S) \in [0,1], \rho_{\phi } \in {\cal T}({\cal H}), S \in {\cal P}({\cal H})quantum state mapping projections to numbers in [0,1] in a sigma-additive way
f \in L^{\infty}(X,\mu) – essentially bounded measurable functions (bounded random variables) A \in {\cal B}({\cal H}) – von Neumann algebra of bounded operators (bounded observables)
\int fg\ d\mu, g \in L^1(X,\mu) – expectation value of f \in L^{\infty}(X,\mu) with respect to p_g

\text{Tr}(\rho A), \rho \in {\cal T}({\cal H}) – expectation value of A \in {\cal B}({\cal H}) in state \rho

In summary, the fact that measurements don’t always commute lead us to consider non-commutative operator algebras. This leads us to the Hilbert space representation of quantum mechanics where a quantum state is a trace-one density operator and an observable is a bounded linear operator. We also saw that projections can be viewed as 0-1 events. The spectral theorem is used to decompose operators into a sum or integral of projections.

The richer mathematical setting for quantum mechanics allows us to model non-classical phenomena such as quantum interference and entanglement. We have not mentioned the time evolution of states, but in short, state vectors evolve unitarily according to the Schrödinger equation, generated by an operator known as the Hamiltonian.

References and Further Reading

[1] Hall, B.C., Quantum Theory for Mathematicians, Springer, Graduate Texts in Mathematics #267, June 2013 (relevant section)

[2] Redei, M., Von Neumann’s work on Hilbert space quantum mechanics

[3] Blackadar, B., Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

[4] Wilce, Alexander, “Quantum Logic and Probability Theory“, The Stanford Encyclopedia of Philosophy (Spring 2017 Edition), Edward N. Zalta (ed.).

[5] Wikipedia – Quantum logic

[6] – Lattice of Projections

[7] – Spectral Measure

[8] quantum mechanics – Intuitive meaning of Hilbert Space formalism – Physics Stack Exchange

[9] This answer to: mathematical physics – Quantum mechanics in a metric space rather than in a vector space, possible? – Physics Stack Exchange

[10] functional analysis – Resolution of the identity (basic questions) – Mathematics Stack Exchange


September 28, 2015

First space probes to visit bodies of the Solar System

Filed under: science — ckrao @ 11:11 am

The table below shows the first space probes to visit various bodies of the Solar System (and the year) by mission type (flyby, orbit, impact or soft landing). Notably 2015 had three events: MESSENGER ended its four year orbit of Mercury with the first impact on the planet, Dawn became the first spacecraft to orbit a dwarf planet (Ceres, in the asteroid belt), and New Horizons flew by Pluto.


flyby orbit impact soft landing
Sun Helios 2 1976 (within 43m km) Luna 1 1959
Mercury Mariner 10 1974 MESSENGER 2011 MESSENGER 2015
Venus Venera 1 1961 Venera 9 1975 Venera 3 1966 Venera 9 1975
Mars Mariner 4 1965 Mariner 9 1971 Mars 2 1971 Mars 3 1971
Jupiter Pioneer 10 1973 Galileo 1995 Galileo 1995
Saturn Pioneer 11 1979 Cassini 2004
Uranus Voyager 2 1986
Neptune Voyager 2 1989
Pluto New Horizons 2015
Ceres Dawn 2015
Moon Luna 1 1959 Luna 10 1966 Luna 2 1959 Luna 9 1966
Titan Huygens 2005
asteroid Galileo asteroid 951 Gaspra – 1991 NEAR Shoemaker asteroid 433 Eros – 2000 NEAR Shoemaker asteroid 433 Eros – 2000
comet ICE comet Giacobini-Zinner – 1985 Rosetta comet Churyumov-Gerasimenko – 2014 Deep Impact – Impactor comet Tempel -2005 Philae comet Churyumov-Gersimenko – 2014




July 30, 2015

Nineteenth century non-avian dinosaur discoveries

Filed under: nature,science — ckrao @ 11:05 am

Below is an attempted chronological list of non-avian dinosaur discoveries of the 19th century that today are considered valid genera. There may still be some where there are only scant remains of the fossil (e.g. a tooth or single bone remain). The list came from [1] with some help from [2] and Wikipedia to filter out doubtful names. Many of the best known dinosaurs are listed here and it looks like most of the major groups are covered. Good histories of dinosaur paleontology are in [3] and [4].


Genus Discoverer Year Dinosaur type
Megalosaurus Buckland 1824 tetanuran (stiff-tailed) theropod
Iguanodon Mantell 1825 beaked ornithopod
Streptospondylus von Meyer 1830 megalosaurid
Hylaeosaurus Mantell 1833 armoured
Thecodontosaurus Riley & Stutchbury 1836 prosauropod
Plateosaurus von Meyer 1837 prosauropod
Poekilopleuron Eudes-Deslongchamps 1838 megalosaurid
Cardiodon Owen 1841 sauropod
Cetiosaurus Owen 1841 sauropod
Pelorosaurus Mantell 1850 brachiosaur
Aepisaurus Gervais 1852 sauropod
Oplosaurus Gervais 1852 sauropod
Massospondylus Owen 1854 prosauropod
Nuthetes Owen 1854 maniraptoran
Troodon Leidy 1856 raptor
Stenopelix von Meyer 1857 pachycephalosaur
Astrodon Johnston 1858 sauropod
Hadrosaurus Leidy 1858 duckbilled ornithopod
Compsognathus J. A. Wagner 1859 coelurosaur (fuzzy theropod)
Scelidosaurus Owen 1859 armoured
Echinodon Owen 1861 heterodontosaurid (early bird-hipped dinosaur)
Polacanthus Owen vide [Anonymous] 1865 armoured
Calamospondylus Fox 1866 oviraptorosaur
Euskelosaurus Huxley 1866 prosauropod
Acanthopholis Huxley 1867 armoured
Hypselosaurus Matheron 1869 sauropod
Hypsilophodon Huxley 1869 beaked ornithopod
Rhabdodon Matheron 1869 beaked ornithopod
Ornithopsis Seeley 1870 sauropod
Struthiosaurus Bunzel 1870 armoured
Craterosaurus Seeley 1874 stegosaurian
Chondrosteosaurus Owen 1876 sauropod
Macrurosaurus Seeley 1876 sauropod
Allosaurus Marsh 1877 carnosaur
Apatosaurus Marsh 1877 sauropod
Camarasaurus Cope 1877 sauropod
Dryptosaurus Marsh 1877 tyrannosaur
Dystrophaeus Cope 1877 sauropod
Nanosaurus Marsh 1877 early bird-hipped dinosaur
Stegosaurus Marsh 1877 plated dinosaur
Diplodocus Marsh 1878 sauropod
Brontosaurus Marsh 1879 sauropod
Anoplosaurus Seeley 1879 armoured
Coelurus Marsh 1879 coelurosaur (fuzzy theropod)
Mochlodon Seeley 1881 beaked ornithopod
Craspedodon Dollo 1883 horned dinosaur
Ceratosaurus Marsh 1884 theropod
Anchisaurus Marsh 1885 prosauropod
Camptosaurus Marsh 1885 beaked ornithopod
Aristosuchus Seeley 1887 coelurosaur (fuzzy theropod)
Ornithodesmus Seeley 1887 raptor
Cumnoria Seeley 1888 beaked ornithopod
Priconodon Marsh 1888 armoured
Coelophysis Cope 1889 early theropod
Nodosaurus Marsh 1889 armoured
Triceratops Marsh 1889 horned dinosaur
Barosaurus Marsh 1890 sauropod
Claosaurus Marsh 1890 duckbilled ornithopod
Ornithomimus Marsh 1890 ostrich dinosaur
Ammosaurus Marsh 1891 prosauropod
Torosaurus Marsh 1891 horned dinosaur
Argyrosaurus Lydekker 1893 sauropod
Sarcolestes Lydekker 1893 armoured
Dryosaurus Marsh 1894 beaked ornithopod


[1] Dinosaur Genera List –

[2] Genus List for Holtz (2007) Dinosaurs

[3] Benton, M. J. 2000. A brief history of dinosaur paleontology. Pp. 10-44, in Paul, G. S. (ed.), The Scientific American book of dinosaurs. St Martin’s Press, New York. –

[4] Equatorial Minnesota The generic history of dinosaur paleontology 1699 to 1869 –

September 18, 2013

Spacecraft missions to outer planets, minor planets or comets

Filed under: science — ckrao @ 11:50 am

After recently hearing news about Voyager 1 being the first human-made object to leave our solar system, I looked up other spacecraft that have visited the outer planets, minor planets (e.g. asteroids) and comets.

Outer Planets

 Spacecraft  Launch Date  Agency  Remarks
Pioneer 10 3-Mar-72 NASA / ARC flyby of Jupiter in Dec 1973, lost contact in Jan 2003 when 12b km from earth
Pioneer 11 6-Apr-73  NASA /ARC  flyby of Jupiter in Dec 1974, Saturn in May 1979, lost contact in 1995
Voyager 2 20-Aug-77  NASA / JPL flyby of all outer planets, the only one to fly by Uranus and Neptune
Voyager 1 5-Sep-77  NASA / JPL flyby of Jupiter in Sep 1977, Saturn in Mar 1979, now 19b km from earth
Galileo 18-Oct-89 NASA flyby of Venus, Earth, asteroids Gasra and Ida (both incidental) before detailed study of Jupiter and its moons, probe dropped into its atmosphere
Ulysses 6-Oct-90 NASA / ESA flyby of Jupiter (Feb 1992), orbiting sun, measuring solar wind and gamma ray bursts
Cassini–Huygens 15-Oct-97 NASA/ESA flyby of Jupiter & Saturn, Huygens deployed from Cassini, landed on Saturn’s moon Titan in Jan 2005
New Horizons 19-Jan-06  NASA expected to fly by Pluto in July 2015
Juno 5-Aug-11  NASA expected to reach Jupiter in Aug 2016

Minor Planets and comets (other than Halley)

 Spacecraft  Launch Date  Agency  Remarks
Dawn 27-Sep-07 NASA orbited Vesta July ’11-Sep ’12, expected to reach Ceres in Feb ’15
Hayabusa 9-May-03 JAXA visited asteroid Itokawa bringing back to earth tiny grains of asteroidal material in June 2010
NEAR Shoemaker 17-Feb-96 NASA studied near-earth asteroid Eros, touched down in Feb 2001
Deep Space 1 24-Oct-98 NASA/JPL flybys of asteroid Braille and Comet Borelly
Stardust 7-Feb-99 NASA/JPL returned dust samples from Comet Wild 2, also intercepted comet Tempel 1 in Feb 2011
Deep Impact 12-Jan-05 NASA/JPL impactor collided with nucleus of comet Tempel in July 2005, extended mission EPOXI flew by Comet 103P/Hartley, now on its way to asteroid 2002 GT
Rosetta 2-Mar-04 ESA flybys of asteroids Steins and Lutetia in 2008, 2010; expected to reach comet 67P/Churyumov-Gerasimenko in mid 2014 deploying the lander Philae
Chang’e-2 1-Oct-10 CNSA flew by asteroid 4179 Toutatis in Dec 2012 after orbiting moon
International Cometary Explorer (ICE) 12-Aug-78 NASA/ESA first spacecraft to pass through comet tail (Comet Giacobini-Zinner)

Halley’s Comet (during its most recent near-earth encounter in 1985-6)

 Spacecraft  Launch date  Agency  Remarks
Giotto 2-Jul-85 ESA passed within 600km of Halley’s comet, then flew by Comet Grigg-Skjellerup in Jul 1992
Vega 1 15-Dec-84 USSR flyby of Venus (descent craft surfaced on Venus) and Halley’s Comet (within 9,000km Mar ’86)
Vega 2 21-Dec-84 USSR flyby of Venus (descent craft surfaced on Venus) and Halley’s Comet (Mar ’86)
Suisei 18-Aug-85 ISAS (now part of JAXA) within 151,000km of Comet Halley in Mar ’86
Sakigake 7-Jan-85 ISAS (now part of JAXA) within 7m km of Comet Halley

Further reading:

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