Quantum Mechanics for Gravitation

Quantum mechanics enters gravitation before any candidate theory of quantum gravity does. It supplies the language of modes, rotations, controlled approximations, barrier transmission, resonances, entanglement, and reduced descriptions. This chapter builds that language as a bounded bridge to quantum fields on flat and curved backgrounds. It is not a compressed replacement for a full course in quantum mechanics, quantum field theory, or quantum gravity.

Helpful background. Calculus and Linear Algebra supplies complex vector spaces, inner products, eigenvalues, and basis changes.

Conventions. This chapter restores $\hbar$ so that spectra, phases, and semiclassical control parameters remain visible. We retain $c=k_{\mathrm B}=1$ and keep Newton’s constant $G$ explicit whenever gravity enters.

Enter the chapter

The six pages form a deliberate sequence. The first three establish the kinematics, dynamics, and symmetry of an isolated quantum system. The next three explain how to approximate that dynamics, describe subsystems, and impose the boundary conditions that distinguish bound states from scattering states and resonances.

Page	Central question	Result carried forward
Quantum States, Observables, and Measurement	Which parts of a prediction refer to preparation, dynamics, and measurement?	Hilbert-space states, self-adjoint observables, Born probabilities, and conditional update
Unitary Dynamics and Pictures of Motion	How can states evolve in one picture while observables evolve in another?	Unitary propagation, conserved norm, Heisenberg evolution, and the exact interaction picture
Angular Momentum, Spin, and Rotations	Why does one algebra organize both orbital modes and intrinsic spin?	Rotation generators, $SU(2)$ representations, angular-momentum addition, and spherical harmonics
Perturbation Theory, Adiabatic Evolution, and WKB	Which dimensionless quantity controls each approximation?	Separate small parameters, leading formulas, and explicit failure conditions
Density Operators, Entanglement, and Reduced States	How can a subsystem be mixed when the total state is pure?	Partial traces, Schmidt spectra, purification, and no-signalling checks
Scattering, Bound States, and Tunneling	How do boundary conditions and poles distinguish spectral classes?	Flux-normalized transmission, partial waves, resonances, and barrier penetration

Readers meeting quantum mechanics for the first time should follow the order shown. This overview itself needs only the helpful background listed above, but the first page uses linear evolution and boundary-value reasoning; read Ordinary Differential Equations first if those tools are unfamiliar. Readers who already know the formalism can use the final column to find the convention or calculation needed for their next topic.

A prediction separates preparation, dynamics, and measurement

A density operator $\rho$ describes a preparation: it is positive, has unit trace, and may represent either a pure state or a statistical mixture. A unitary operator $U$ propagates that state through the interval of interest. For a projective measurement, outcome $a$ is represented by an orthogonal projector $P_a$, with $P_a^2=P_a=P_a^\dagger$ and $\sum_aP_a=I$. The probability of that outcome is

$$p(a)=\operatorname{Tr}\!\left(P_aU\rho U^\dagger\right).$$

Each factor has a different operational role. Changing $\rho$ changes the preparation, changing $U$ changes the intervening dynamics, and changing $P_a$ changes the question asked at the end. The formula gives outcome probabilities; a separate conditional-update rule describes the selected post-measurement state. Preskill develops the projective rule and its density-operator form in Preskill 2015, §2.1, Eqs. (2.5)–(2.12), printed pp. 4–7, and §2.3, Eq. (2.57), printed p. 18.

A two-level check makes the roles concrete. Let $|0\rangle$ and $|1\rangle$ be an orthonormal basis. The Pauli operator $\sigma_x=|0\rangle\langle1|+|1\rangle\langle0|$ swaps the two basis states. Prepare $\rho=|0\rangle\langle0|$, evolve with

$$U(\theta) =\cos\!\frac{\theta}{2}\,I -i\sin\!\frac{\theta}{2}\,\sigma_x,$$

and test the projector $P_1=|1\rangle\langle1|$. Then

$$p(1)=\operatorname{Tr}\!\left(P_1U\rho U^\dagger\right) =\sin^2\!\frac{\theta}{2}.$$

At $\theta=0$ the preparation is recovered; at $\theta=\pi$ the population is fully transferred. The calculation uses no interpretation-dependent shortcut: it is a direct preparation–dynamics–measurement prediction.

Five structures reappear in gravitational physics

The chapter is selective because later gravitational calculations reuse a small number of quantum structures again and again.

Structure	What it controls here	Later gravitational use	Limit of the transfer
Self-adjoint generators and unitary evolution	Norm conservation and equivalent pictures of motion	Mode evolution and interaction-picture perturbation theory	Time-dependent geometry or tracing out degrees of freedom need not give a closed unitary subsystem evolution
Rotation representations	Orbital harmonics, spin, and addition rules	Scalar, spinor, vector, and tensor mode decompositions	The field equations and gauge constraints still determine which modes are physical
Controlled approximations	Perturbative shifts, slow driving, and WKB phases	High-frequency propagation, level mixing, and barrier estimates	A small parameter must be stated; turning points, degeneracies, and gap closings need separate treatment
Reduced states	Local predictions from a composite state	Entanglement across regions and effective open-system descriptions	A reduced density operator depends on a declared subsystem split
Scattering boundary conditions	Flux, bound states, tunneling, and continued poles	Greybody transmission and the resonance grammar behind quasinormal modes	Black-hole asymptotics, superradiance, spin, and coupled channels modify the elementary potential problem

The unity comes from operators and boundary data, not from pretending that all five rows are the same approximation. Schrödinger, Heisenberg, and interaction pictures are exactly equivalent before truncation. WKB and perturbation theory are approximations with distinct control parameters. A partial trace is an exact mathematical operation once the tensor-product split has been chosen. A resonance is defined by analytic continuation and outgoing conditions, not by the visual appearance of a damped oscillation.

The bridge stops before quantum field theory and quantum gravity

Fixed-particle quantum mechanics cannot create or annihilate quanta, define a local relativistic field algebra, choose a vacuum on a curved spacetime, or compute stress–energy backreaction. Those tasks require the Flat-Spacetime Quantum Field Theory for Gravitation chapter and, later, quantum field theory in curved spacetime and semiclassical gravity.

Even the phrase “subsystem” needs care once gauge constraints and gravity are present. In perturbation theory about flat spacetime for $D\gt3$, begin with a compactly supported quantum-field-theory operator $O$ that transforms nontrivially under Poincaré transformations. Making it gauge invariant requires a gravitationally dressed operator $\widetilde O$ whose leading correction depends on the asymptotic metric; the dressed observable is therefore not compactly supported. Donnelly and Giddings establish this restricted obstruction in Donnelly and Giddings 2016, Introduction and §III, printed pp. 2–10. It is not a proof that useful approximate subsystems never exist, nor is it a signalling theorem. The density-operator page therefore teaches the ordinary tensor-product model and states explicitly where later gravitational theory must refine it.

Review the chapter

Use this as an exit check after the reduced-states page. For a bipartite density operator $\rho_{AB}$ and any orthonormal basis $\{|b\rangle\}$ of subsystem $B$, define

$$\operatorname{Tr}_B\rho_{AB} =\sum_b (I_A\otimes\langle b|)\rho_{AB}(I_A\otimes|b\rangle).$$

The von Neumann entropy is $S(\rho)=-\operatorname{Tr}(\rho\ln\rho)$.

For the bipartite state

$$|\Phi^+\rangle =\frac{|00\rangle+|11\rangle}{\sqrt2},$$

find the reduced state of subsystem $A$ and compare its entropy with the entropy of the complete state. Then show that applying any unitary $U_B$ only to subsystem $B$ cannot change $\rho_A$.

Solution

The total density operator is $\rho_{AB}=|\Phi^+\rangle\langle\Phi^+|$. Taking the partial trace over $B$ removes the cross terms because $\langle0|1\rangle=0$:

$$\rho_A =\operatorname{Tr}_B\rho_{AB} =\frac12\bigl(|0\rangle\langle0|+|1\rangle\langle1|\bigr) =\frac{I_A}{2}.$$

Therefore $S(\rho_A)=\ln2$, while the complete state is pure and has $S(\rho_{AB})=0$. The local mixedness records entanglement across the chosen partition; it does not turn the global preparation into a classical mixture.

After a unitary on $B$,

$$\rho'_{AB} =(I_A\otimes U_B)\rho_{AB}(I_A\otimes U_B^\dagger).$$

Evaluate the partial trace in an orthonormal basis $\{|b\rangle\}$ of $B$:

$$\begin{aligned} \rho'_A &=\sum_b (I_A\otimes\langle b|U_B)\rho_{AB} (I_A\otimes U_B^\dagger|b\rangle)\\ &=\operatorname{Tr}_B\rho_{AB} =\rho_A, \end{aligned}$$

because $\{U_B^\dagger|b\rangle\}$ is also an orthonormal basis. This is the unitary special case of the no-signalling check developed on the reduced-states page.

References

• Donnelly, William, and Steven B. Giddings. “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems.” Physical Review D 94 (2016): 104038. doi:10.1103/PhysRevD.94.104038. Open PDF.
• Preskill, John. Quantum Information, Chapter 2, revised July 2015, §2.1, printed pp. 3–7; §§2.3–2.5, printed pp. 16–35. California Institute of Technology lecture notes. Official PDF.