The indistinguishability of particles is one of the hallmarks distinguishing classical from quantum systems, and also a concept difficult for beginners to understand. However, textbooks never discuss the conditions for using second quantization methods and the validity of particle indistinguishability, with rather arbitrary choices in specific applications. This article will discuss these issues and help everyone better understand why quantum mechanics is an approximate theory.
Second Quantization
Second quantization is a mathematical method for expressing many-particle quantum systems. It introduces particle creation and annihilation operators, mapping systems with variable particle numbers into Fock space. Second quantization treats particles as field excitations: bosons through commutators [â^i, â^j†] = δ_ij, and fermions through anticommutators {ĉ^i, ĉ^j†} = δ_ij.
Why do we need second quantization? In many-particle systems, first quantization (standard quantum mechanics) struggles to handle particle number changes (such as particle creation/annihilation processes). Second quantization provides a unified framework that naturally describes these dynamic processes. For example, in quantum field theory, it allows us to handle relativistic effects and particle creation, while in condensed matter physics, it simplifies many-body interaction calculations. Second quantization is not "quantizing classical systems a second time," but rather a second representation of already quantized systems to accommodate many-particle statistics and symmetry requirements.
From the perspective of canonical quantization, first quantization promotes classical mechanical variables (like coordinates and momentum) to operators and introduces wave functions to describe system states. This is a process of "wave-ification" or "functionalization," transforming particle trajectories into probability waves. Second quantization goes further with "functionalization": it treats the wave function itself as a field operator, allowing particle number as a variable. This extends the system from fixed-particle-number Hilbert space to variable-particle-number Fock space, making it more efficient to handle phenomena like superconductivity or Bose-Einstein condensation.
Comparison with First Quantization
First and second quantization are not opposing but complementary. Once quantized, we choose tools within the mathematical framework: first quantization suits fixed-particle-number systems, describing wave function evolution in Hilbert space; second quantization targets variable-particle-number or identical particle systems, using creation/annihilation operators to construct Fock space. Their differences include:
Particle number: First quantization assumes fixed particle numbers (like electrons in hydrogen atoms), while second quantization allows particle number changes (like photon generation in photon fields).
Symmetry: First quantization requires manually imposing exchange symmetry (symmetric/antisymmetric wave functions), while second quantization automatically implements Bose/Fermi statistics through operator commutation relations.
Computational convenience: Second quantization simplifies many-body Hamiltonian expressions, with Hamiltonians directly expressed using ĉ†ĉ terms for transitions.
Non-relativistic quantum mechanics' second quantization (as applied in many-body physics) is similar but not identical to quantum field theory's second quantization. The former is typically non-relativistic, focusing on condensed matter systems (like electron gas), ignoring particle creation; the latter is relativistic, satisfying Lorentz invariance, used in particle physics. Both share mathematical structure, but the quantum field theory version is more general, capable of handling vacuum excitations and antiparticles.
Hilbert space versus Fock space comparison: Hilbert space is a complete inner product space for single-particle or fixed multi-particle systems, supporting linear superposition and spectral decomposition. For many-particle systems, Hilbert space is the tensor product of single-particle Hilbert spaces, requiring handling of boson or fermion symmetry issues. Fock space, however, is the direct sum of Hilbert tensor products: F = ⊕_{N=0}^∞ H^⊗N, allowing different particle number subspaces. Fock space is better suited for identical many-particle systems, naturally incorporating statistical symmetry, while Hilbert space requires additional handling.
Are Identical Particles Actually Distinguishable or Indistinguishable?
Identical particles divide into bosons and fermions: bosons have symmetric collective wave functions, allowing identical quantum state occupation; fermions have antisymmetric collective wave functions, obeying the Pauli exclusion principle. This distinction is very significant and fundamental.
How do we determine if particles are distinguishable/indistinguishable? "Identical" by definition means indistinguishable, but what about in practice? Identical particles mean many-particle systems cannot individually distinguish particles, cannot number particles, with overall wave functions symmetric or antisymmetric. The question is: where's the boundary? It's impossible for all similar particles in the universe (like all electrons) to form a single global state.
For bosons, taking helium atoms as an example: Their ground state is bosonic (spin 0), with electrons and nuclei both satisfying Bose statistics. According to Bose statistics, at low temperatures, helium atoms should spatially condense together. But at low temperatures, liquid helium-4 forms Bose-Einstein condensate (BEC) without all atoms "degenerating together" into overlapping states. This is because BEC as a macroscopic quantum phenomenon is also affected by temperature and interactions, not requiring all identical particles to globally condense. Van der Waals forces between helium atoms prevent complete overlap, showing that indistinguishability isn't absolute but depends on system properties and conditions. For boson systems, indistinguishability is only significant when interactions can be neglected.
Moreover, valence electrons and core electrons have overlapping wave function spaces but are often distinguished in many-body calculations (like Hartree-Fock methods) because their energies and orbitals differ. The widely applied Density Functional Theory (DFT) doesn't consider electron indistinguishability at all, instead using density to describe average behavior. DFT ignores exchange correlation to simplify calculations, yet is very successful in materials science.
Contrary to bosons, the "quantum nature" of fermion systems is not only not weakened by interactions but amplified under strong interactions. In the non-interacting limit, fermions only exhibit the Pauli exclusion principle, forming Fermi surfaces and degeneracy pressure. When interactions are introduced, phenomena emerge such as: Mott insulators, superconductors (BCS pairing), topological insulators, quantum Hall states, strongly correlated systems (like heavy fermions, non-Fermi liquids), etc. Fermion indistinguishability is the foundation of all collective behavior, especially prominent in strong coupling.
Generally, spatial separation is a condition for treating particles as separable and independent. But should spatial separation always mean independent particles? Not necessarily. According to general understanding, two identical particles in quantum entanglement (like Bell states) remain indistinguishable even when spatially separated, because their wave functions are entangled. Using the entanglement concept, spatially separated particles can be described as "Bell particles," maintaining indistinguishability.
However, in quantum computing, quantum bits (qubits) often satisfy the identical definition (like identical spin-1/2 particles), at least mathematically. But quantum computing doesn't use the concept of identical particles, let alone second quantization, instead using classical labeling (like "qubit 1" and "qubit 2"). Whether from the perspective of identical particles or entanglement, this labeling method seems to violate the fundamental principles that quantum computing researchers advocate.
These examples show that in different applications, people's treatment of quantum indistinguishability is quite arbitrary, not following "orthodox" symmetry and statistics.
Conditions and Limitations of Identical Particles
Identical particles are only phenomena under ideal global conditions: many-particle systems exhibit indistinguishability due to symmetry. Under strongly correlated conditions, like high-energy, close-distance collisions in quantum field theory, (fermion) identical effects are more prominent because inter-particle exchange paths cannot be ignored. Conversely, low-energy identical effects (like BEC) often require low temperatures to reduce interaction effects.
These are natural in global approximate interpretations: quantum mechanics assumes ideal wave propagation (infinite-speed action), ignoring actual propagation finiteness. The conditions required for indistinguishability are more stringent, requiring global coherent coordination of all particles (like zero-temperature limit), thus more easily violated. Actually, even first quantization (canonical quantization) requires conditions: it assumes the system is isolated, ignores relativistic effects, wave nature dominates, and reaching eigenstates requires no time.
Textbooks don't discuss conditions, making choices appear arbitrary, but they're actually understandable. For example, DFT ignores indistinguishability yet is effective because in low-density solids, electrons are approximately independent; quantum computing uses classical labeling because qubits are well-isolated (then how to ensure mutual entanglement, might there be issues here?), not requiring consideration of exchange symmetry.
Quantum Mechanics as an Approximate Theory
The treatment choices for second quantization and identical particles show that quantum mechanics' mathematical expression isn't a universal principle but an approximate framework, dependent on global ideal conditions like infinite propagation speed and system isolation. When these conditions clearly don't hold, we return to classical physics. Mathematics is rigorous, but the model itself can be crude. Of course, under ideal conditions, the model can also be very precise.