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Observable

Observable is a San Francisco-based software company that provides a cloud-based platform for collaborative data exploration, analysis, and visualization. The platform enables teams to create interactive notebooks integrating code, data queries, and dynamic graphics, supporting languages like , SQL, and while emphasizing real-time collaboration and AI-assisted insights. Founded in 2016 by , the creator of the visualization library, and Melody Meckfessel, a former executive, Observable aims to make data work more accessible and trustworthy for diverse users including developers, analysts, and business professionals. The core of Observable's offering is its reactive computational notebook environment, where cells execute code to generate live outputs such as charts, maps, and dashboards, with changes propagating automatically across the document. Key features include seamless integration with databases like Snowflake and BigQuery, support for open-source libraries like D3 and Plot, and tools for embedding visualizations in external applications or exporting them as static files. The platform's design draws from Bostock's expertise in web-based graphics, evolving from an initial focus on JavaScript-centric exploration to a broader ecosystem that includes the Observable Framework for building data apps and the Observable Canvases for AI-enhanced analysis. Since its public launch in 2018, Observable has grown through community contributions, with over 1.2 million notebooks created. Observable has secured significant venture funding to expand its capabilities, including a $10.5 million Series A round in November 2020 led by and Acrew Capital and a $35.6 million Series B in led by Menlo Ventures, bringing total funding to $46.1 million. As of 2025, the company reports milestones such as over 508 million downloads of , 350,000 users, and the launch of Observable Canvases, positioning it as a leader in modern data collaboration tools.

Overview

Definition

In physics, an observable is a physical property or quantity whose value can be determined through empirical measurement. Examples include position, momentum, and energy in mechanics, which can be quantified using appropriate instruments under controlled conditions. Unlike abstract theoretical variables—such as parameters in mathematical models that lack direct empirical access—observables are distinguished by their measurability and relevance to experimental verification. This empirical accessibility ensures that observables bridge theoretical predictions with real-world data, forming the foundation of scientific inquiry across disciplines. General examples span various fields: in , serves as a key observable, representing the average of particles and measurable via thermometers or spectroscopic methods. In , is an observable that quantifies motion and can be determined from over time using tools like or optical tracking. In , the measurability of observables implies reproducibility: under identical initial conditions, repeated measurements yield the same value, underscoring the deterministic framework of the theory. This property contrasts with quantum contexts, where observables are associated with operators, though their measurement outcomes exhibit probabilistic characteristics.

Historical Development

The concept of observables in physics originated in the foundations of during the 17th century. advanced the empirical measurement of quantities such as position, velocity, and time through his inclined-plane experiments on falling objects, establishing a quantitative approach to motion that emphasized observable data over qualitative speculation. This work directly influenced Isaac Newton's Philosophiæ Naturalis Principia Mathematica (1687), which integrated these observables into a comprehensive deterministic system governed by laws of motion and universal gravitation, allowing precise predictions from initial conditions. By the 19th century, the deterministic nature of classical observables reached its philosophical zenith with Pierre-Simon Laplace's articulation of causal determinism. In his 1814 Essai philosophique sur les probabilités, Laplace described a hypothetical superintelligence—later termed Laplace's demon—that could compute the future state of the universe entirely from the positions and momenta of all particles at one moment, embodying the classical ideal of fully predictable observables. This view reinforced the belief that physical quantities like position and momentum were inherently deterministic and accessible through complete measurement. The early marked a profound transition as anomalies in atomic phenomena exposed limitations in classical observables, leading to the quantum revolution. Niels Bohr's principle of complementarity, presented in his 1927 Como lecture, shifted focus from underlying hidden variables to the inherent incompatibility of certain observables in quantum measurements, resolving paradoxes by prioritizing empirical outcomes over classical . Pivotal milestones defined this quantum framework: Werner Heisenberg's 1925 paper introduced , representing observables as non-commuting arrays to model discrete quantum transitions and spectral lines, diverging from continuous classical variables. John von Neumann's 1932 Mathematical Foundations of Quantum Mechanics further solidified the shift by formalizing observables within , transforming them from definite classical parameters into probabilistic operators whose measurements yield eigenvalue spectra with inherent uncertainties. Post-1920s thus supplanted Laplace's deterministic worldview with a probabilistic paradigm, where observables reflect statistical predictions rather than absolute certainties.

Classical Physics

Observables in Classical Mechanics

In , observables are real-valued smooth functions defined on the , a that encodes the complete state of a through its position coordinates q and conjugate momentum coordinates p. The is typically represented as the T^*M over the configuration space manifold M, where each point (q, p) specifies the system's configuration and its generalized momenta. This formalism allows observables to capture measurable physical quantities, such as or , as functions A: T^*M \to \mathbb{R}. Common examples of observables include the T = \frac{p^2}{2m}, which depends solely on the for a particle of m; the V(q), a of alone; and the , embodied by the H = T + V, which governs the system's . These functions are evaluated at specific points in to yield numerical values for the observables at a given state. The itself serves as the primary observable for and in isolated systems. The dynamics of observables are described using the Poisson bracket, a bilinear operation that encodes the symplectic structure of phase space. For two observables A and B in a system with n degrees of freedom, the Poisson bracket is defined as \{A, B\} = \sum_{i=1}^n \left( \frac{\partial A}{\partial q_i} \frac{\partial B}{\partial p_i} - \frac{\partial A}{\partial p_i} \frac{\partial B}{\partial q_i} \right). This bracket satisfies antisymmetry (\{A, B\} = -\{B, A\}), bilinearity, and the Jacobi identity, endowing the space of observables with a Lie algebra structure. The time evolution of an observable A follows Hamilton's equation \frac{dA}{dt} = \{A, H\} + \frac{\partial A}{\partial t}, where the partial derivative accounts for any explicit time dependence in A; if A is time-independent and \{A, H\} = 0, then A is a conserved quantity. In one dimension, the bracket simplifies to \{A, B\} = \frac{\partial A}{\partial q} \frac{\partial B}{\partial p} - \frac{\partial A}{\partial p} \frac{\partial B}{\partial q}. Measurement in is inherently deterministic: given ideal instruments, the exact values of all observables can be obtained simultaneously for a in a well-defined state (q, p), as there are no limitations on or compatibility between measurements. This contrasts with , where observables become non-commuting operators on a .

Measurement and Determinism

In classical physics, the measurement of observables entails direct empirical determination of their numerical values under controlled conditions, typically using instruments such as rulers to gauge position or clocks to record time. This process assumes negligible back-action from the measurement apparatus on macroscopic systems, allowing observables like velocity or energy to be ascertained with high precision without significantly altering the system's state. Such measurements form the foundation for empirical validation of physical laws, where the observable's value is extracted reproducibly from the system's interaction with the measuring device. The of underpins the interpretation of these , asserting that the future evolution of a classical is uniquely determined by its initial state. formalized this in , describing an intellect that, knowing the positions and momenta (key observables) of all particles at time t=0, could predict the entire future trajectory through Newton's laws. This Laplacian implies that perfect of observables eliminates , rendering the 's path predictable in , as the are reversible and time-symmetric. However, practical limitations arise in chaotic systems, where infinitesimal errors in measuring initial observables are exponentially amplified over time, leading to divergence in predicted trajectories. The exemplifies this sensitivity, as small uncertainties in positions or velocities render long-term forecasts infeasible despite the underlying deterministic equations. Theoretically, these challenges are resolvable with arbitrarily precise measurements, preserving , though computational and observational constraints impose effective unpredictability. In , observables are often treated as ensemble averages over many realizations of the system, bridging microscopic measurements to macroscopic properties. For instance, in an emerges as the time-averaged per unit area from molecular collisions, yielding the equation of state PV = NkT, where P is the observable , V the volume, N the number of particles, k Boltzmann's , and T the . This averaging approach accounts for the collective behavior of unmeasurable individual particle s, providing a deterministic framework for thermodynamic predictions.

Quantum Mechanics

Observables as Operators

In , observables are mathematically represented by (Hermitian) operators acting on the of the quantum system. This formulation, introduced by , ensures that the possible outcomes of measurements correspond to the real eigenvalues of these s, reflecting the empirical reality that measurement results are real numbers. Self-adjointness guarantees that the is equal to its , \hat{A}^\dagger = \hat{A}, which is essential for the probabilistic interpretation of quantum states. Specific examples illustrate this operator representation. In the position basis, the position operator \hat{x} is defined as multiplication by the position coordinate x, so \hat{x} \psi(x) = x \psi(x) for a wave function \psi(x). The momentum operator \hat{p}, in the same representation, takes the differential form \hat{p} = -i \hbar \frac{d}{dx}, where \hbar is the reduced Planck's constant. These canonical operators satisfy the commutation relation [\hat{x}, \hat{p}] = i \hbar, foundational to the Heisenberg uncertainty principle, though their explicit forms depend on the chosen basis. The general structure of an observable \hat{A} is provided by the for operators, which allows a \hat{A} = \sum_i \lambda_i |\psi_i \rangle \langle \psi_i |, where \{\lambda_i\} are the distinct eigenvalues representing possible outcomes, and |\psi_i \rangle are the corresponding orthonormal eigenvectors forming a complete basis of the . For continuous spectra, the sum generalizes to an over the spectral measure. This decomposition underpins the probabilistic nature of measurements, as the probability of obtaining eigenvalue \lambda_i in |\psi \rangle is |\langle \psi_i | \psi \rangle|^2. The expectation value of an observable \hat{A} in a normalized state |\psi \rangle is given by the inner product \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle, which represents the average outcome over many measurements on identically prepared systems. Using the spectral decomposition, this simplifies to \langle \hat{A} \rangle = \sum_i \lambda_i |\langle \psi_i | \psi \rangle|^2, weighting each eigenvalue by its probability. This operator framework distinguishes quantum observables from their classical counterparts, where they are simply real-valued functions on phase space.

Compatible Observables

In , two observables represented by operators  and B̂ are compatible if they commute, meaning their vanishes: [ \hat{A}, \hat{B} ] = \hat{A}\hat{B} - \hat{B}\hat{A} = 0. This commutation relation implies that both observables can be measured simultaneously with arbitrary precision, without one measurement disturbing the outcome of the other. The concept originates from the foundational axiomatization of , where observables are associated with linear operators on , and compatibility ensures that the measurement process preserves the eigenstates relevant to both. A key mathematical property of compatible observables is that commuting self-adjoint operators share a common complete set of eigenstates. If |\psi\rangle is an eigenstate of  with eigenvalue \lambda, so \hat{A} |\psi\rangle = \lambda |\psi\rangle, then |\psi\rangle is also an eigenstate of B̂ with some eigenvalue \mu, satisfying \hat{B} |\psi\rangle = \mu |\psi\rangle. This shared eigenbasis arises from the simultaneous diagonalizability of commuting operators, as guaranteed by the in theory, allowing the observables to be jointly diagonalized. Representative examples of compatible observables include the Ĥ (total energy) and components of in systems governed by central potentials, such as the potential in the . Here, Ĥ commutes with the square of the \hat{L}^2 and its z-component \hat{L}_z, i.e., [ \hat{H}, \hat{L}^2 ] = 0 and [ \hat{H}, \hat{L}_z ] = 0, enabling the specification of energy levels alongside angular momentum quantum numbers without conflict. Another example is the in one spatial direction and the in an orthogonal direction, such as \hat{x} and \hat{p}_y, which commute since [ \hat{x}, \hat{p}_y ] = 0, reflecting the independence of perpendicular coordinates in free space./02%3A_Introduction_to_Quantum_Mechanics/2.05%3A_Operators_Commutators_and_Uncertainty_Principle) The implications of extend to the prediction of joint outcomes. For a |\psi\rangle, the joint for obtaining eigenvalues \lambda of  and \mu of B̂ is P(\lambda, \mu) = |\langle \phi_{\lambda \mu} | \psi \rangle|^2, where |\phi_{\lambda \mu}\rangle denotes the common eigenstates labeled by both eigenvalues. This distribution fully characterizes the simultaneous statistics, underscoring how resolves the otherwise indeterminate correlations between observables in .

Incompatible Observables

In , two observables represented by Hermitian operators  and B̂ are defined as incompatible if their [Â, B̂] ≠ 0. This non-commutativity implies that the operators do not share a complete set of common eigenstates, preventing the existence of a eigenbasis and thereby prohibiting the precise simultaneous determination of definite values for both observables in any . A example of incompatible observables is the x̂ and the p̂, which satisfy the commutation relation [x̂, p̂] = iℏ. This relation underpins the duality between the and representations of the wave function, where precise knowledge in one domain inherently broadens the uncertainty in the other. The general consequences of incompatibility are profound: no can serve as a simultaneous eigenstate of both operators, and a of one observable inevitably disturbs the system's preparation with respect to the other. Heisenberg illustrated this disturbance through his 1927 gamma-ray microscope , in which high-resolution imaging of an electron's using short-wavelength gamma rays causes unpredictable transfers via , thus altering the electron's trajectory. This notion of incompatible observables was pivotal in the philosophical debates between Niels Bohr and Albert Einstein at the Fifth Solvay Conference in 1927, where discussions centered on whether quantum mechanics fully captures the objective reality of physical observables or introduces fundamental indeterminacy. Such incompatibility also gives rise to uncertainty relations that quantify the trade-offs in measurement precision.

Measurement and Interpretation

Eigenvalue Measurements

In quantum mechanics, the measurement of an observable \hat{A} is governed by the projection postulate, which specifies both the possible outcomes and their probabilities. For a system in a normalized state |\psi\rangle, the observable \hat{A} possesses a complete set of orthonormal eigenstates |i\rangle with corresponding eigenvalues \lambda_i. Upon measurement, the system yields one of the eigenvalues \lambda_i with probability |\langle i | \psi \rangle|^2, the square of the modulus of the projection of |\psi\rangle onto the eigenstate |i\rangle. This probabilistic nature arises because the state |\psi\rangle is generally a superposition of the eigenstates, and the measurement selects one eigenvalue according to the Born rule. Following the measurement outcome \lambda_i, the wave function undergoes an instantaneous collapse to the corresponding eigenstate |i\rangle, a process known as the von Neumann projection. This non-unitary projection abruptly reduces the superposition to a definite eigenstate, marking the transition from quantum superposition to a classical-like definite outcome. The postulate, formalized by John von Neumann in his 1932 treatise, ensures that the post-measurement state is normalized and aligned with the measured eigenvalue. A key consequence of this collapse is the repeatability of measurements: if the same observable \hat{A} is measured immediately after on the collapsed state |i\rangle, the outcome \lambda_i is obtained with certainty (probability 1), as |i\rangle is an eigenstate. This property underscores the definitive nature of the measurement process and distinguishes it from the unitary evolution governing the system between measurements. A illustrative example is the measurement of the z-component of spin, S_z, for a spin-1/2 particle, where the eigenvalues are \pm \hbar/2 with eigenstates |+\rangle and |-\rangle. For a general state |\psi\rangle = \cos(\theta/2) |+\rangle + \sin(\theta/2) e^{i\phi} |-\rangle, represented on the by a point at polar angle \theta from the +z axis, the probability of measuring +\hbar/2 is \cos^2(\theta/2), and -\hbar/2 is \sin^2(\theta/2). Post-measurement, the state collapses to either |+\rangle or |-\rangle, ensuring a subsequent S_z measurement yields the same result deterministically.

Expectation Values and Uncertainty

In , the expectation value of an observable represented by a Hermitian \hat{A} provides the average result of repeated measurements on an ensemble of systems prepared in the state |\psi\rangle. If |\psi\rangle = \sum_i c_i |\lambda_i\rangle in the eigenbasis of \hat{A}, with eigenvalues \lambda_i and probabilities |c_i|^2, then the expectation value is given by \langle \hat{A} \rangle = \sum_i \lambda_i |c_i|^2. This quantity, introduced in the foundational formalism of , connects the probabilistic nature of measurements to a value analogous to classical statistics. The time dependence of expectation values follows from the dynamics of the wave function, as encapsulated in the . For a general observable \hat{A}, possibly time-dependent, the theorem states \frac{d\langle \hat{A} \rangle}{dt} = \left\langle \frac{\partial \hat{A}}{\partial t} \right\rangle + \frac{i}{\hbar} \langle [\hat{H}, \hat{A}] \rangle, where \hat{H} is the and [\cdot, \cdot] denotes the . This relation demonstrates how quantum expectation values approximate classical trajectories for sharply peaked states, bridging the two theories. To characterize the fluctuations around this average, the variance of an observable \hat{A} is defined as \Delta A^2 = \langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2, where \Delta A measures the standard deviation or intrinsic uncertainty in measurement outcomes. For pairs of incompatible observables \hat{A} and \hat{B} that do not commute, these uncertainties obey the Robertson–Schrödinger relation: \Delta A \Delta B \geq \frac{1}{2} \left| \langle [\hat{A}, \hat{B}] \rangle \right|. Originally derived by Howard P. Robertson in 1929 as a general inequality from the Cauchy-Schwarz relation applied to states, it was refined by Erwin Schrödinger in 1930 to include covariance terms, emphasizing the role of non-commutativity in limiting simultaneous precision. A prominent illustration occurs for position \hat{x} and momentum \hat{p}, where [\hat{x}, \hat{p}] = i\hbar yields \Delta x \Delta p \geq \hbar/2. Equality holds for Gaussian wave packets, which minimize the product of uncertainties and represent coherent states with balanced spreads in . These packets, evolving under free-particle dynamics, exhibit spreading over time while preserving the minimum-uncertainty property at t=0.

Extensions and Applications

In Quantum Field Theory

In (QFT), observables are extended to relativistic settings through field operators defined over . For a real , the primary observable is the field operator \phi(x), which acts as an operator-valued distribution on the Hilbert space of the theory, smeared with test functions to ensure well-definedness. The conjugate is \pi(x) = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi)}, where \mathcal{L} is the , typically \mathcal{L} = \frac{1}{2} \partial^\mu \phi \partial_\mu \phi - \frac{1}{2} m^2 \phi^2 for a massive . These operators generalize the and operators from non-relativistic to an infinite-dimensional field space. The algebraic structure of these observables is governed by canonical commutation relations imposed during quantization. At equal times, [\phi(\mathbf{x}, t), \pi(\mathbf{y}, t)] = i \hbar \delta^3(\mathbf{x} - \mathbf{y}), while [\phi(\mathbf{x}, t), \phi(\mathbf{y}, t)] = [\pi(\mathbf{x}, t), \pi(\mathbf{y}, t)] = 0. These relations extend the familiar [x, p] = i \hbar from single-particle quantum mechanics to the field theory context, ensuring the correct Poisson bracket promotion to quantum operators across the spatial degrees of freedom. In the full relativistic framework, the commutation relations are extended to all spacetime points, with fields at spacelike separations satisfying [\phi(x), \phi(y)] = 0 when (x - y)^2 < 0. Vacuum expectation values of these field operators provide essential correlation functions that encode the theory's dynamics. The vacuum state |0\rangle is annihilated by all destruction operators, and expectation values like \langle 0 | \hat{A} | 0 \rangle for a observable \hat{A} yield vacuum condensates or . A key example is the two-point \langle 0 | \phi(x) \phi(y) | 0 \rangle = \int \frac{d^4 p}{(2\pi)^4} \frac{i}{p^2 - m^2 + i\epsilon} e^{-i p \cdot (x - y)} in , which represents the and is fundamental for computing scattering amplitudes and Green's functions in . A central challenge in defining QFT observables is ensuring locality and , which require that measurements in spacelike-separated regions do not influence each other instantaneously. This is enforced by the microcausality condition: observables associated with spacelike-separated regions must commute, [O(x), O(y)] = 0 for (x - y)^2 < 0, preventing superluminal signaling and aligning the theory with . Violations of this principle would undermine the of , making it a foundational in axiomatic formulations of QFT.

In Quantum Information Theory

In quantum information theory, observables play a central role in characterizing the properties of , which are the basic units of encoded in two-dimensional Hilbert spaces. The Pauli operators—\sigma_x, \sigma_y, and \sigma_z—serve as the set of observables for a single , each being Hermitian with eigenvalues \pm 1. These operators correspond to measurements along the x, y, and z directions in the representation of the state. For instance, measuring \sigma_z projects the onto the computational basis states |0\rangle or |1\rangle, yielding outcomes +1 or -1 with probabilities determined by the state's \rho, specifically p(+1) = \langle 0 | \rho | 0 \rangle and p(-1) = \langle 1 | \rho | 1 \rangle. Similar projections occur for the other Pauli bases, enabling the extraction of superposition and phase information essential for quantum computation. For entangled qubits, observables reveal non-local correlations that violate classical bounds, as exemplified by Bell observables in the Clauser-Horne-Shimony-Holt (CHSH) inequality. This inequality involves incompatible spin measurements on two entangled particles, such as combinations of \sigma_z and rotated Pauli operators, where the expectation value of the Bell operator B = \sigma_z \otimes \sigma_z + \sigma_z \otimes \sigma_x + \sigma_x \otimes \sigma_z - \sigma_x \otimes \sigma_x satisfies | \langle B \rangle | \leq 2 under local hidden variable theories. Quantum mechanics predicts a maximum violation of $2\sqrt{2} \approx 2.828 for the singlet state, demonstrating entanglement through these incompatible observables. Experimental confirmation came in 1982, where time-varying analyzers measured photon polarizations, achieving a violation exceeding the classical limit by 5 standard deviations, thus ruling out local realism. Information-theoretic measures quantify the uncertainty and information content associated with observables in quantum systems. The von Neumann entropy S(\rho) = -\Tr(\rho \log \rho) of the density operator \rho measures the intrinsic uncertainty of the quantum state, providing an upper bound on the classical information accessible via measurements of observables. For a general observable \hat{A} implemented via a positive operator-valued measure (POVM), the Shannon entropy of the measurement outcomes quantifies the extracted classical information, subject to bounds such as the Holevo bound. POVMs allow generalized measurements beyond projective ones to optimize information extraction while preserving quantum coherence in applications like quantum cryptography. A key application of observables in is , which reconstructs the full \rho from statistics of measurements on an informationally complete set of observables, such as the Pauli basis for qubits. By estimating expectation values \langle \hat{O}_i \rangle = \operatorname{Tr}(\rho \hat{O}_i) for multiple observables \hat{O}_i via repeated preparations and measurements, linear inversion or maximum-likelihood methods yield \rho, enabling verification of quantum states and computation of arbitrary observables. This technique is vital for benchmarking quantum devices, with efficient protocols reducing the number of required measurements to scale favorably for low-dimensional systems.