**Paris** | **21 June 2018**

Bryan W. Roberts

London School of Economics

philsci-archive.pitt.edu/14449

Primitive labels

Real-number labels

Real-plane labels

Complex-plane labels

**Negative numbers:** Initial Skepticism

**Negative numbers** can represent the world

**Imaginary numbers:** Initial Skepticism

**Imaginary numbers** can represent the world

"There is no representation except in the sense that some things are used, made, or taken, to represent some things as thus and so."

— van Fraassen 2008, *Scientific Representation*, pg.23

**Some things** represented by complex numbers 'thus and so'

- Classical waves
- Oscillating current
- Polynomial roots

**MEANWHILE,** in modern quantum theory...

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

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

"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

**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

**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)

**Thesis:**

The **standard dogma** that observables must be self-adjoint (with real spectrum) **should be relaxed**.

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

Observables: **a Disassembly Theorem**

Observables: **a Disassembly Theorem**

Observables: **a Disassembly Theorem**

Observables: **a Disassembly Theorem**

**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:

These depend on **eigenvectors**, not eigenvalues.

...and **expectation values** as usual,

As expected, it is a **complex number**!

**Why this works.** Normal operators have a spectral resolution, which guarantees well-behaved probabilities.

For \(A\) **self-adjoint** (\(A=A^*)\),

the spectral theorem implies well-behaved probabilities:

For \(A\) **normal** (\(AA^*=A^*A)\),

the spectral theorem still holds:

"All... operators do not possess a complete, orthonormal set of eigenfunctions. However, theHermitian operators capable of representing physical quantitiespossess 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 | \(iQ, \; i\sigma_z\) |

unitary | \(e^{-itH}\) |

other functions | \(f(Q), \; g(\sigma_z)\) |

\(\vdots\) | \(\vdots\) |

**Heterodoxy**

"I am happy forthe 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 thatmy quantum 'observables' be normal linear operators"

— Roger Penrose (2004), *Road to Reality*

**Heterodoxy**

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"

It turns out that **not all normal operators** can be considered observables at once.

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

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

**Unobservables in quantum theory**

**Unobservables in quantum theory**

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

\(\sigma_x\sigma_y\) or 'Joint spin-\(x\) 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 interpreted** as 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 \([A,B] = 0\).

"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 \(i\sigma_z = \sigma_x\sigma_y\).

...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?

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

**extension** of orthodox Quantum Theory

**Adjoint**of an operator \(A\) is an operator \(A^*\) such that \( \langle \psi, A\phi \rangle = \langle A^*\psi,\phi\rangle\) for all \(\psi,\phi\in D_A\).**Symmetric**operator \(A\) is one such that \( \langle \psi, A\phi \rangle = \langle A\psi,\phi\rangle\) for all \(\psi,\phi\in D_A\).**Self-adjoint**operator \(A\) is a symmetric operator such that \(D_A = D_{A^*}\).

**Example: Time observables** 'track time' in a dynamical theory

- Classical mechanics: \(\tau(s_t) = \tau(s_0)+t\)

**Time observables** 'track time' in a dynamical theory

- Schrödinger QM: \(\mathrm{Tr}(T\rho_t) = \mathrm{Tr}(T\rho_0) + t\)
- Heisenberg QM: \(T(t) = T(0) + tI\)

**Why 'track time': physics reason.**

Predicting the occurrence of a jet

**Why 'track time': philosophy reason.**

We have measuring devices for it

**Why 'track time': philosophy reason.**

We have measuring devices for it

**Why 'track time': philosophy reason.**

We have measuring devices for it

Energy **bounded from below**

Energy **bounded from below**

Energy **unbounded from below** (implausible)

**Pauli's Theorem.**

Let \(H\) be a self-adjoint operator ('energy') that is bounded from below.

If \(T\) is self-adjoint, then it is not a time observable.

**Pauli's Theorem (Contrapositive).**

Let \(H\) be a self-adjoint operator ('energy') that is bounded from below.

If \(T\) is a time observable, then it 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)

**Classical Example:** The free particle

- \(h(q,p) = p^2/2m,\;\; \tau(q,p) = mq/p\)
- \(\Rightarrow \{h,\tau\}=1\).

**Quantum Example:** The free particle

**Time observable:**\([H,T]=i\) when \(H=P^2/2m\) <
**Symmetric,**obviously**Not self-adjoint**: \(D_T \neq D_{T^*}\)

**Naimark spectral theorem.** Every symmetric operator that \(A\) that is closed and densely defined admits a POVM \(\Delta\mapsto E_\Delta\) such that \(A=\int_\mathbb{R}\lambda dE_\lambda\). (It is a PVM iff \(A\) is self-adjoint.)

(Dubin and Hennings 1990, *Quantum mechanics, algebras and distributions*)

**3. Real spectrum** Observables:

**extension** of orthodox Quantum Theory

**Example:** The operator,

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)

philsci-archive.pitt.edu/12478/

Thank you.

Momentum generates **spatial translation**

Angular momentum generates **spatial rotation**

In orthodox quantum mechanics these symmetries \(U_a\) are strongly continuous 1-parameter unitary representations of \((\mathbb{R},+)\).

**Stone's theorem:** Then \(U_a = e^{isA}\) and \(A\) is self-adjoint.

Stone's theorem \(\Rightarrow\) **observables are self-adjoint?**

- Observables
**can**generate symmetries, but there is no a priori reason that they**must**. - \(iA\) is the generator of \(U_a = e^{s(iA)}\). Why promote the part that is self-adjoint?

- If \(U_a\) is a strongly continuous 1-parameter unitary semigroup, then \(U_a = e^{iaB}\), and the generator \(B\) is normal.
- If \(B\) is normal, then \(U_a = e^{iaB}\) is a strongly-continuous 1-parameter semigroup.

**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-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

**Continuous Symmetry**

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

**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?

No.

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

strictly speaking.

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