Unreal Observables



Bryan W. Roberts
London School of Economics

MCMP | 14 June 2017

Some Inspiration

"Working in various physics departments for a couple of years, I had the chance to attend several PhD examinations. Usually, after the candidate derived a wanted result formally on the blackboard, one of the members of the committee would stand up and ask: 'But what does it mean? How can we understand that \(x\) is so large, that \(y\) does not contribute, or that \(z\) happens at all?' Students who are not able to tell a 'handwaving' story in this situation are not considered to be good physicists."

Stephan Hartmann (1999)
"Models and stories in hadron physics"

Overheard in Many PhD Examinations:

"Since it is a physical observable, it must be represented by a self-adjoint operator"




Primitive labels


Real-number labels


Real-plane labels


Complex-plane labels

Negative numbers: Initial Skepticism

negative numbers

Negative numbers can represent the world

negative numbers

Imaginary numbers: Initial Skepticism

complex numbers

Imaginary numbers can represent the world

complex numbers

And we do use complex numbers to represent things.

  • Classical waves
  • Oscillating current
  • Fourier theory
  • Kähler manifolds

MEANWHILE, in quantum mechanics...

Textbook dogma

"Now, the expectation value of an observable quantity has got to be a real number (after all, it corresponds to actual measurements in the laboratory, using rulers and clocks and meters)"

Griffiths (1995), Introduction to Quantum Mechanics, §3.3

Textbook dogma

"We expect on physical grounds that an observable has real eigenvalues.... That is why we talk about Hermitian observables in quantum mechanics."

Sakurai (1994), Modern Quantum Mechanics, §1.3

Philosophical tradition

Identifying the textbook dogma...

"the standard textbook way [to interpret quantum theory] is to associate measurements with certain self-adjoint operators" — Wallace (2012) The Emergent Multiverse

Philosophical tradition

Following the textbook dogma...

"it's clear... (since, of course, the values of physically measurable quantities are always real numbers) that the operators associated with measurable properties must necessarily be Hermitian operators."

Albert (1992) Quantum mechanics and experience

"The analogy with classical [Fourier] theory leads further to allowing as representatives of real quantities only those matrices that are 'Hermitian'" — Born and Heisenberg, Report for the 1927 Solvay Conference

Born and Heisenberg, Report for the 1927 Solvay Conference, drawing on Born, Heisenberg and Jordan (1925) "QM II"

Classical Harmonic Analysis
\(\psi(x,t) = A\cos(z)\), with \(z := \tfrac{2\pi}{\lambda}(x - vt).\)

Convenient for calculations: \(Ae^{iz} = A\cos(z) + iA\sin(z)\)

To make predictions: \(\mathrm{Re}\left(Ae^{iz}\right) = A\cos(z)\)

"These assumptions are reasonable on account of the eigenvalues of real [Hermitian] linear operators being always real numbers." — (Dirac 1930, Quantum Mechanics, pg. 35 of 3rd/4th edition)



The standard dogma that observables must be self-adjoint should be relaxed.

The reward: New interpretive questions, new symmetries, and new physics.

Observables: a Disassembly Theorem

observable flower \[ A^*\psi = A\psi, \;\;\; D_A = D_{A^*} \]

Observables: a Disassembly Theorem

observable flower \[ A^*\psi = A\psi \]

Observables: a Disassembly Theorem

observable flower \[\mathrm{eigenvalues}(A)\subseteq\mathbb{R}\]

Observables: a Disassembly Theorem

observable flower \[ AA^*=A^*A \]
observable flower

Eigenvalue Equation

\[ \sigma\varphi = a\varphi. \]

Eigenvector \(\varphi\) of an matrix \(\sigma\) with eigenvalue \(a\):

Self-adjoint spin observable \(\sigma_z = \bigl(\begin{smallmatrix}1 & \\ & -1 \end{smallmatrix}\bigr)\)

Eigenvectors \(\varphi^\uparrow,\varphi^\downarrow\), Eigenvalues \(\pm 1\)

Non-self-adjoint spin observable \(i\sigma_z = \bigl(\begin{smallmatrix}i & \\ & -i \end{smallmatrix}\bigr)\)

Same Eigenvectors \(\varphi^\uparrow,\varphi^\downarrow\), Eigenvalues: \(\pm i\)

Call \(i\sigma_z\) an unreal observable.

Can we use it to do physics?

We can assign probabilities as usual:

\[\Pr\textstyle_\psi(\varphi^\uparrow) = |\langle \varphi^\uparrow, \psi \rangle|^2 \]

These depend on eigenvectors, not eigenvalues.

...and expectation values as usual,

\[ \langle\psi,A\psi\rangle = i\Pr\textstyle_\psi(\varphi^\uparrow) + (-i)\Pr\textstyle_\psi(\varphi^\downarrow) \]

As expected, it is a complex number!

Why this works

Probabilities are well-behaved for all normal operators

There are two properties underpinning this.

Property 1. A normal operator is equivalent to a pair of 'simultaneously measurable' self-adjoints.

  • We can always write \(A = B + iC\) with \(B,C\) self-adjoint
  • Exercise: \(A\) is normal iff \([B,C]=0\)
  • Then \(B\) and \(C\) have a common spectral resolution, i.e. are 'simultaneously measurable'

Property 2. Normal operators have a spectral resolution, which guarantees well-behaved probabilities.

For \(A\) self-adjoint (\(A=A^*)\),
the spectral theorem implies well-behaved probabilities:

\[ \Pr\textstyle_\psi(\varphi_1) + \Pr\textstyle_\psi(\varphi_2) + \cdots + \Pr\textstyle_\psi(\varphi_n) = 1 \]

For \(A\) normal (\(AA^*=A^*A)\),
the spectral theorem still holds:

\[ \Pr\textstyle_\psi(\varphi_1) + \Pr\textstyle_\psi(\varphi_2) + \cdots + \Pr\textstyle_\psi(\varphi_n) = 1 \]
"All... operators do not possess a complete, orthonormal set of eigenfunctions. However, the Hermitian operators capable of representing physical quantities possess such a set. For this reason we give the name 'observable' to such operators."

Messiah (1961) Quantum Mechanics (Vol 1, Ch.V.9)

  • But the same applies to normal operators too!


Normal operators (\(AA^*=A^*A\)) can be observables.

self-adjoint\(Q, \; \sigma_z\)
anti-Hermitian\(iQ, \; i\sigma_z\)
other functions\(f(Q), \; g(\sigma_z)\)

Interpreters of QM have a lot more freedom in how they apply mathematics.

The free Klein-Gordon field \(\phi(x)\) is a normal, non-self-adjoint field operator. It can be viewed as a pair of self-adjoint observables.

On this perspective, it is also an observable in its own right.


Jordan's early (1926) formalism for matrix mechanics suggests treating normal operators as observables.

Duncan and Janssen (2013),
"(Never) Mind your p's and q's"


"I am happy for the results of measurements (eigenvalues) to be complex numbers, while insisting on the standard requirement of orthogonality between the alternative states that can result from a measurement, I shall demand only that my quantum 'observables' be normal linear operators"

— Roger Penrose (2004), Road to Reality

Sharp sets

Observables: \(\sigma_x, \sigma_y, \sigma_z\)

Observables: \(i\sigma_x, i\sigma_y, i\sigma_z\)

Unobservables in quantum theory

\(|\uparrow\rangle_x = \tfrac{1}{\sqrt{2}}(|\uparrow\rangle_y + |\downarrow\rangle_y)\)

\(\sigma_x\sigma_y\) or 'Joint spin-\(z\) and spin-\(y\)' is unobservable

Unobservables in quantum theory

Fact: \(\sigma_x\sigma_y = i\sigma_z\).

So, if \(\sigma_x\) and \(\sigma_y\) are observables,
then \(i\sigma_z\) is unobservable!

But \(i\sigma_z\) can still be an observable...

Just not at the same time as \(\sigma_x\) and \(\sigma_y\)


Definition. A sharp set of operators \(S\) is one such that if \(A,B,AB\in S\), then \([B,C] = 0\).

"A sharp set contains no products of incompatibles"

  • Such products are unobservables in quantum theory.
  • \(S = \{\sigma_x, \sigma_y, \sigma_z\}\) is a sharp set.
  • \(S' = \{i\sigma_x, i\sigma_y, i\sigma_z\}\) is a sharp set too.
  • \(S\cup S'\) is not a sharp set, since \(i\sigma_z = \sigma_x\sigma_y\).

What I know about sharp sets

...and more things that I want to know!

Proposition 1. The normal operators in \(B(\mathcal{H})\) are not a sharp set.

Proposition 2. Every set of self-adjoint operators is sharp.

Proposition 3. The set \(SA\) of all self-adjoint operators is not a maximal sharp set.

Why? Because \(SA\cup\{cI \;|\; c\in\mathbb{C}\}\) is sharp.

Implications for Observability: There may be more observables than we think there are.

  • Q1: What is the maximal Bell correlation for sharp sets of normal operators? (Is it different from \(2\sqrt{2}\)?)
  • Q2: Can 'observables' like AB-phase or Berry-phase be modeled using sharp sets?
  • Q3: What else are we missing?

Proposition 4. If \(\mathcal{S}\) is a sharp set, then for any linear operator \(A\), the set \(A\mathcal{S}A^{-1}\) is also sharp.

Implications for Symmetry: There are symmetries in quantum theory than we thought there were (and in particular non-unitary symmetries).

  • Q4: Is a transformation a physical symmetry if it maps sharp sets to sharp sets?
  • Q5: Can the sharp sets in the set of normal operators in \(B(\mathcal{H})\) be classified?

Q6: What is an example of a maximal sharp set?

(Is \(SA\cup Z(SA)\) maximal, where \(Z(SA)\) is the 'center' of the set SA?)

What I don't know could fill volumes...

2. Symmetric Observables (\(A\psi=A^*\psi\)):

observable flower

extension of orthodox Quantum Theory

Time observables 'track time' in that,
\(\langle \psi_t, T\psi_t\rangle = \langle \psi_0, T\psi_0\rangle + t\).

  • Of physical, philosophical interest: see Pashby (2014).
  • They are not self-adjoint, but are symmetric.

Pauli's Theorem. Let \(H\) be self-adjoint and be bounded from below, and let \(U_t = e^{-itH}\). If \(T\) is a time observable, then it is not self-adjoint.

Fact: A huge class of physical systems admit symmetric, non-self-adjoint time observables. (Muga et al 2011, Pashby 2014)

3. Real spectrum Observables:

observable flower

extension of orthodox Quantum Theory

Example: The operator,

\[ \begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix} \]

has only real eigenvalues \(\lambda=1,2\)

with eigenvectors \(\bigl(\begin{smallmatrix}1 \\ 1 \end{smallmatrix}\bigr)\) and \(\bigl(\begin{smallmatrix}1 \\ 0 \end{smallmatrix}\bigr)\).

But it is not self-adjoint.

Example: The operator,


\[H = \tfrac{1}{2m}P^2 + Q^2 + iQ^3\]


has an entirely real spectrum.

But it is not self-adjoint.

PT-Symmetric Quantum Mechanics explores techniques for constructing such operators, and is of interest in supersymmetric extensions of QM.

(Bender and Boettcher 2000, Dorey et al. 2001)




Euan Squires (1988) `Non-self-adjoint observables':

"faster-than-light signalling is an inevitable consequence of the assumption that 'measurements'... of non-self-adjoint operators can be made"
No, it isn't. (See Barnett and Kraemer Phys. Lett. A 2002)

No-Signaling is Very General

Let \(\mathcal{A},\mathcal{B}\) be subalgebras of \(\mathcal{M}\) with \([\mathcal{A},\mathcal{B}]=0\). If \(E_i\in\mathcal{A}\) satisfy \(\sum E_i = I\), and,

\[T(A) := \sum_{i=1}^n E_i^{1/2}AE_i^{1/2},\]

then for all states \(\omega:\mathcal{M}\rightarrow\mathbb{C}\) and all \(B\in\mathcal{B}\),

\[ (T^*\omega)(B) = \omega(T(B)) = \omega(B). \]

Summary: For commuting systems, 'measuring' one doesn't affect the other

Symmetries and Normal Operators


Momentum generates spatial translation



Angular momentum generates spatial rotation

Continuous Symmetry
(\(U_t\) a strongly continuous 1-parameter unitary rep)

\[ \Updownarrow \]

Self-Adjoint Generator
(\(U_t = e^{itH}\) with \(H\) self-adjoint and conserved)

Does this mean observables are self-adjoint, since only the generator of a symmetry is an observable?


1. Anti-Hermitian generator \(iH\) for \(U_t = e^{itH}\),
strictly speaking.

2. Non-self-adjoint operators are conserved too, including \(iH, e^{iH}, f(H), \dots\)


observable flower



observable flower