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Contents

Idea

The Born rule is the core statement of coordination in the foundations of quantum physics:

In its version for pure states the Born rule says that

• for $A$ an observable in the form of a Hermitean operator on the given Hilbert space $\mathcal{H}$ of pure quantum states with Hermitian inner product $\langle\text{-}\vert\text{-}\rangle$

• $P$ denoting the projection operator on a given eigenstate of this operator,

• $\pi \,\in\, \mathcal{H}$ a pure state;

then the probability of observing the given eigenstate in a quantum system which is in state $\psi$ equals (in bra-ket notation):

In particular, if $W$ is an orthonormal basis for the Hilbert space of states associated with a quantum measurement-procedure, then the probability of measuring the result $w$ on a system in state $\left\vert \psi \right\rangle$ is

and so if the state is normalized to begin with ($\langle \psi \vert \psi \rangle$ = 1 ), then the probability is

Essentially in this form the rule was first formulated by Born 1926b p 805.

Related concepts

• quantum probability

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

Historical origins

The Born rule is named in honor of

• Max Born, Zur Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 37 (1926) 863–867 [[doi:10.1007/BF01397477](https://doi.org/10.1007/BF01397477)]

where it appears (though disregarding the norm symbol) as a brief footnote-added-in-proof:

\begin{imagefromfile} “file_name”: “Born_stating_the_BornRule-Footnote.jpg”, “width”: 500, 560, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

and its expanded version

• Max Born, Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 38 (1926) 803–827 [[doi:10.1007/BF01397184](https://doi.org/10.1007/BF01397184)]

which provides the mathematical details (p. 805):

\begin{imagefromfile} “file_name”: “Born_stating_the_BornRule-Expanded.jpg”, “width”: 460, 480, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

and finally in the followup

• Max Born, Das Adiabatenprinzip in der Quantenmechanik, Zeitschrift für Physik 40 (1927) 167–192 [[doi:10.1007/BF01400360](https://doi.org/10.1007/BF01400360)]

which makes the interpretation more explicit (p. 171):

\begin{imagefromfile} “file_name”: “Born_stating_the_BornRule-ExpandedII.jpg”, “width”: 500, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

Early The adoption generalization of the this Born idea rule: to probabilitydensities for continuous observables is due to Wolfgang Pauli, as first recounted in:

• Pascual Jordan, Über eine neue Begründung der Quantenmechanik, Zeitschrift für Physik 40 (1927) 809–838 [[doi:10.1007/BF01390903](https://doi.org/10.1007/BF01390903)]

\begin{imagefromfile} “file_name”: “Jordan-PauliBornRule.jpg”, “width”: 510, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

and then by Pauli himself, again in a footnote:

• Wolfgang Pauli, Über Gasentartung und Paramagnetismus, Zeitschrift für Physik: A Hadrons and nuclei 41 (1927) 81–102 [[doi:10.1007/BF01391920](https://doi.org/10.1007/BF01391920)]

\begin{imagefromfile} “file_name”: “PauliStatingBornPauliRule.jpg”, “width”: 540, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

Early review of the Born-Pauli rule:

• David Hilbert, John von Neumann, Lothar W. Nordheim, Über die Grundlagen der Quantenmechanik, Math. Ann. 98 (1928) 1–30 [[doi:10.1007/BF01451579](https://doi.org/10.1007/BF01451579)]

where it is once again a footnote:

\begin{imagefromfile} “file_name”: “HilbertNeumannNordheim-BornRule.jpg”, “width”: 500, 530, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

and then in its full recognition and amplification as a pillar of quantum physics:

• Paul A. M. Dirac, The physical interpretation of the quantum dynamics, Proceedings of the Royal Society of London 113 765 (1927) [[doi:10.1098/rspa.1927.0012](https://doi.org/10.1098/rspa.1927.0012)]

\begin{imagefromfile} “file_name”: “Dirac_OnBornPauliRule.jpg”, “width”: 530, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

The full recognition and amplification of the Born rule as a pillar of quantum physics making it a probabilistic theory (cf. quantum probability) is (maybe besides Jordan 1927) due to:

• John von Neumann, Die quantenmechanische Statistik, Part III in:

  Mathematische Grundlagen der Quantenmechanik, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  Mathematical Foundations of Quantum Mechanics Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), Wikipedia entry]

\begin{imagefromfile} “file_name”: “vonNeumann-BornRule.jpg”, “width”: 500, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

\begin{imagefromfile} “file_name”: “vonNeumann-QMStatistics.jpg”, “width”: 500, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}

Historical survey:

• Jagdish Mehra, Helmut Rechenberg, The Probability Interpretation and the Statistical Transformation Theory, the Physical Interpretation, and the Empirical and Mathematical Foundations of Quantum Mechanics 1926-1932, Part 1 in: The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics, 1926-1941, Springer (2001) [[ISBN:978-0-387-98971-6](https://link.springer.com/book/9780387989716)]

Further discussion

Review:

• Klaas Landsman, The Born rule and its interpretation, in: Compendium of Quantum Physics, Springer (2009) 64-70 [[doi:10.1007/978-3-540-70626-7_20](https://doi.org/10.1007/978-3-540-70626-7_20), pdf, pdf]

See also:

• Wikipedia, Born rule