philsci-archive.pitt.edu/12964/

Bryan W. Roberts

London School of Economics

MCMP | 14 June 2017

"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"

"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** can represent the world

**Imaginary numbers:** Initial Skepticism

**Imaginary numbers** can represent the world

**And we do** use complex numbers to represent things.

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

**MEANWHILE,** in quantum mechanics...

"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)

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**

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**!

Probabilities are well-behaved for all **normal operators**

(\(AA^*=A^*A\)).

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:

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

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

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**.

**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"

**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*

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

...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 (\(A\psi=A^*\psi\)):

**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:

**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/

**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

Momentum generates **spatial translation**

Angular momentum generates **spatial rotation**

**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 \(iH, e^{iH}, f(H), \dots\)