What is Hilbert Space?

Hilbert space is a complete inner product space, developed by German mathematician David Hilbert and others, primarily used to describe infinite-dimensional or finite-dimensional vector spaces, with wide applications in quantum mechanics, functional analysis, and other fields. Its core definition includes the following elements:

Vector space: Hilbert space is a vector space containing a set of "vectors" (which can be functions, wave functions, etc.), supporting addition and scalar multiplication.

Inner product: Defines an inner product operation to measure the "angle" and "length" between vectors, satisfying properties of positive definiteness, symmetry, and linearity.

Completeness: Hilbert space is complete, meaning any Cauchy sequence converges to a point within the space, ensuring all limit operations are meaningful.

Dimension: Hilbert space can be finite-dimensional (like Euclidean space) or infinite-dimensional (like function spaces).

In quantum mechanics, elements of Hilbert space are typically wave functions describing the system's quantum states, while the inner product is used to calculate probabilities or expectation values. The quantization process transforms classical physics variables (like position, momentum) into operators and state vectors in quantum mechanics. These operators and state vectors need to operate in an appropriate mathematical space, and Hilbert space provides such an environment:

• Hilbert space's linear properties naturally support quantum state superposition (e.g., $|\psi ⟩=c_1|{\varphi }_1⟩+c_2|{\varphi }_2⟩$)

• Inner product supports probability calculations (e.g., $|⟨\varphi |\psi ⟩{|}^2$ represents measurement probability)

• Completeness ensures mathematical rigor (all Cauchy sequences converge)

The "Custom-Made" Tool for Quantum Mechanics

Hilbert space's structure naturally accommodates quantum mechanics' requirements:

Observables and self-adjoint operators: Self-adjoint (Hermitian) operators correspond to observables (position, momentum, energy), with their eigenvalues and eigenvectors directly describing measurement results and possible states, perfectly consistent with quantum mechanics' measurement postulates.

Infinite-dimensional systems: For continuous variable systems (position, momentum), Hilbert space (like $L^2$ space, square-integrable function space) can handle wave functions' continuous spectra, suitable for describing physical quantities like position and momentum.

The Abstractness of Hilbert Space

Hilbert space is an abstract mathematical concept, easily confusing for beginners. By contrast, Euclidean space is the three-dimensional space of daily experience, very intuitive.

Dimensional Abstractness

Euclidean space: The familiar Euclidean space is typically finite-dimensional (e.g., 2D plane or 3D space), intuitively represented by coordinates like $(x,y,z)$. Its vectors are "visible"—displacement vectors or force vectors can be drawn on paper or imagined in physical space.

Hilbert space: Hilbert space can be infinite-dimensional, especially in quantum mechanics where it describes wave functions (like $L^2$ space). These "vectors" are no longer simple coordinate points but functions (like wave function $\psi (x)$), difficult to intuitively understand.

Nature of Elements

In Euclidean space, vectors are concrete geometric objects with clear physical meaning for length and direction.

In Hilbert space, vectors can be wave functions or state vectors describing probability distributions of quantum systems, not direct physical entities. For example, an electron's spin states $|\uparrow ⟩$ or $|\downarrow ⟩$ aren't "arrows" in space but abstract mathematical objects representing quantum states.

Meaning of Inner Product

Euclidean space's inner product (dot product) corresponds to angles or projections, easily linked to geometric intuition (e.g., $a\cdot b=|a||b|\cos \theta$).

Hilbert space's inner product (like $⟨\psi |\varphi ⟩$) represents the "overlap degree" between quantum states, with its modulus squared $|⟨\psi |\varphi ⟩{|}^2$ corresponding to probability. This probabilistic meaning is more abstract than geometric angles, difficult to directly imagine.

Infinite Dimensions and Completeness

Hilbert space's completeness (all Cauchy sequences converge) is a purely mathematical property ensuring rigor of mathematical operations, but has no direct physical intuition counterpart. In contrast, Euclidean space's finite-dimensional nature is closer to everyday experience.

The Disappearance of the "Quantum" Concept

After quantization is complete and the physical system is embedded in Hilbert space's mathematical framework, the "quantum" concept becomes somewhat "invisible," manifested primarily in mathematical structure rather than as special "quantum" properties. Once quantization is complete, the physical system's description completely transfers to Hilbert space's mathematical framework, involving state vectors (wave functions), operators, and inner products. At this point, quantum mechanics' "quantum" characteristics (wave-particle duality, uncertainty principle) are embedded in mathematical structure without needing to explicitly appear.

Hilbert space is an abstract mathematical framework not exclusive to quantum mechanics (also used in signal processing, functional analysis, etc.). In quantum mechanics, Hilbert space's linear structure, inner product, and completeness sufficiently describe quantum state evolution and measurement.

Post-quantization systems operate primarily through Hilbert space's mathematical language, with quantum mechanics' uniqueness (superposition, entanglement, probabilistic nature) translated into linear algebra and operator theory, the "quantum" concept itself receding:

• Superposition principle: Linear combination of state vectors (e.g., $|\psi ⟩=c_1|{\varphi }_1⟩+c_2|{\varphi }_2⟩$)

• Non-commutativity: Commutation relations between operators (e.g., $[\hat{x},\hat{p}]=i\hslash$), reflecting the uncertainty principle

• Measurement: Self-adjoint operator eigenvalues and projection probabilities $|⟨\varphi |\psi ⟩{|}^2$

• Normalization: Quantum state probabilities sum to 1 ($⟨\psi |\psi ⟩=1$), the core of quantum mechanics' probabilistic interpretation, ensuring wave function modulus squared is probability density

Additionally, other operations in Hilbert space (operator action, time evolution) appear as "ordinary" mathematical operations, with quantum mechanics' specificity mainly implicitly manifested through:

Discretization: Eigenvalues of certain observables (energy, angular momentum) are discrete, arising from self-adjoint operators' spectral properties.

Wave-particle duality: Wave function $\psi (x)$ can describe both wave interference (through superposition) and particle probability distribution (through $|\psi (x){|}^2$).

Entanglement: Multi-particle systems' Hilbert space is tensor product space (e.g., ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$), allowing non-local quantum correlations.

From Abstract to Real: Projection of Quantum States

Quantum states ($\psi (x)$ or $|\psi ⟩$) are abstract mathematical objects requiring projection onto physical space for intuition:

Electron Cloud Distribution

Position operator $\hat{r}$'s eigenvector $|r⟩$ corresponds to point $r=(x,y,z)$ in three-dimensional space. The wave function's projection onto position basis (each point in 3D space) is $\psi (r)=⟨r|\psi ⟩$, with modulus squared $|\psi (r){|}^2$ representing probability density of finding the electron at $r$. This projection "maps" the abstract Hilbert space state to three-dimensional Euclidean space, forming intuitive electron cloud images (like hydrogen atom's s, p, d orbitals).

Spin

In non-relativistic quantum mechanics (in Hilbert space), spin is an artificially defined intrinsic physical quantity without classical counterpart (though the electron's actual spin can be physical). Unlike position or momentum, it cannot directly associate with three-dimensional Euclidean space.

For an electron's spin (spin-1/2 particle), its quantum state is defined in two-dimensional Hilbert space, with basis vectors typically taken as $|\uparrow ⟩$ and $|\downarrow ⟩$ (corresponding to spin component along some direction, e.g., $S_z=\pm \hslash /2$). Any spin state can be represented as $|\psi ⟩=a|\uparrow ⟩+b|\downarrow ⟩$ (where $a,b$ are complex numbers satisfying normalization $|a{|}^2+|b{|}^2=1$).

Spin's physical values come directly from Hilbert space: probability of spin-up $(+\hslash /2)$ is $|a{|}^2$; probability of spin-down $(-\hslash /2)$ is $|b{|}^2$. These probabilities and eigenvalues are calculated directly from Hilbert space's state vectors and operators, without needing "projection" to three-dimensional Euclidean space.

Other Physical Quantities

Momentum, angular momentum, and other physical quantities have corresponding projection operators and algorithms. Besides spin, some physical quantities don't require projection, like energy, charge, and mass. It's particularly important to note that multi-particle system entanglement (like Bell states) is a non-local property in Hilbert space tensor products that cannot be visualized through projection to three-dimensional space.

Limitations of Hilbert Space

Hilbert space is a linear space, thus the properties of quantum states derived from it are necessarily linear, ideally globally coherent, with inherent limitations. Conclusions calculated within Hilbert space cannot be used to negate more fundamental and general discussions, such as Schrödinger's cat (non-globally coherent states), causality (valid in relativity), coherence effective range, etc.

The linear description of Hilbert space is based on primitive non-relativistic isolated system idealized assumptions, ignoring potential nonlinear effects, environmental interactions, or other complex factors that may exist in actual physical systems.

In real systems, the coherence of quantum states is typically affected by decoherence, where system-environment interactions destroy the phase relationships of superposition states. This phenomenon cannot be directly described within the standard Hilbert space framework and requires additional theoretical tools (such as density matrices or open quantum system theory).

Taking the Schrödinger's cat debate as an example, the cat as a whole in everyday concepts (neither a point mass nor a rigid body) does not conform to the fundamental definition of quantum (i.e., wave). In Schrödinger's designed experiment, the cat exhibits no dead-alive wave-like behavior and thus cannot be treated as a quantum object.

The linear framework of Hilbert space is the foundation of quantum mechanics, but it does not directly handle spacetime structure or causality. Conclusions based on Hilbert space (such as entanglement's non-locality) cannot negate relativity's causality, as the latter is a more fundamental physical principle requiring reconciliation within a broader framework (such as quantum field theory).

In Hilbert space, the evolution of quantum states (through unitary operators) is global, theoretically allowing instantaneous correlations (non-locality of entangled states), but this non-locality is limited to discussions within Hilbert space and cannot be extrapolated to general cases where relativity holds.

The effective range of coherence is a property of actual physical systems, involving experimental conditions and environmental interactions, while the linear framework of Hilbert space is an idealized mathematical model. Conclusions about global coherence based on Hilbert space cannot negate the finite nature of coherence ranges in reality, as the latter requires more complex physical and mathematical descriptions. This is why discussions about decoherence were later added additionally.

The Collective Unconscious Regarding Limitations

However, in reality, the limitations of the Hilbert space tool are collectively ignored by the vast majority of quantum physics practitioners. Almost no traditional quantum mechanics textbook discusses these limitations of non-relativistic quantum mechanics. Combined with physics students' already unfamiliar, fearful, or mystified attitude toward concepts like operator algebra, functionals, and Hilbert space, if its limitations aren't emphasized, it easily leads to superstition and resistance to reasonable questioning, thus unconsciously standing in opposition to science, essentially betraying science.

Traditional quantum mechanics textbooks (such as Dirac's Principles of Quantum Mechanics, Landau's Quantum Mechanics, Shankar's Principles of Quantum Mechanics) primarily center on Hilbert space, operator algebra, and wave functions, emphasizing the derivation and application of mathematical tools (like the Schrödinger equation, harmonic oscillators, hydrogen atoms). This content is suitable for teaching as it provides a clear mathematical framework, facilitating students' mastery of specific computational techniques.

Non-relativistic quantum mechanics is the introductory framework of quantum mechanics, assuming isolated systems and non-relativistic spacetime structure. This simplification facilitates teaching but rarely emphasizes the limitations of its applicable range. Textbooks and curriculum design typically prioritize students' mastery of skills for solving specific problems (like energy level calculations, spin matrices) rather than deep exploration of theoretical limitations or philosophical questions. This pragmatic tendency means complex issues like decoherence, relativistic quantum mechanics, or open quantum systems are postponed to more advanced courses (like quantum field theory).

For physics majors, mathematical concepts like operator algebra, functional analysis, and Hilbert space are rather abstract, with a steep learning curve. Many students feel confused or "fearful" when first learning these concepts, as they differ greatly from the intuitive geometry of classical physics (like Euclidean space).

Due to the complex and powerful mathematical structure of Hilbert space, students may view it as "mysterious truth" rather than a tool with limitations. This psychological tendency may lead students to over-rely on the mathematical framework during learning while ignoring the underlying physical assumptions and applicable scope.

Textbooks and courses often focus on "how to calculate" while rarely guiding students to question Hilbert space assumptions (such as linearity, isolated systems, global coherence). This may lead students to mistakenly believe that Hilbert space description is the entirety of quantum mechanics, while ignoring broader physical problems.

Non-relativistic quantum mechanics has been extremely successful in explaining phenomena in atoms, molecules, and solid-state physics, and this success may lead researchers and students to overlook its limitations. The mathematical framework of Hilbert space performs so precisely in microscopic systems that people easily view it as "ultimate truth."

The scientific community generally takes a cautious attitude toward questioning core frameworks (like the applicability of Hilbert space) because these frameworks have been widely validated. Researchers may thus unconsciously avoid questioning, even viewing questioning as "unscientific," thereby departing from the critical spirit of science.

Researchers mistakenly believe that the linear description of Hilbert space encompasses all of quantum mechanics, while ignoring more complex physical problems like decoherence, quantum-classical transitions, and causality. For example, does the legitimacy of the macroscopic superposition state of Schrödinger's cat expressed in Hilbert space require additional assumptions that are ignored?

This misunderstanding leads researchers to avoid thinking about philosophical questions in quantum mechanics (such as the measurement problem, the nature of reality). The essence of science lies in continuous questioning and revision of assumptions. If reasonable questioning is resisted due to "superstition" about Hilbert space, this actually violates the scientific spirit—that is, betraying science.