Why do the mathematical definitions of fermionic creation and annihilation operators—specifically their matrix forms—so crucially depend on subtle sign factors? And do these definitions, as written in standard quantum field theory texts, actually satisfy the famous anticommutation relations? This question gets at the core of how quantum theory encodes the profound difference between fermions and bosons, and it turns out the answer is both subtle and illuminating. Let’s dig deep into the structure of these operators, the logic of their construction, and the role of phase factors in ensuring their defining algebra holds true.
Short answer: The matrix definitions of fermionic creation and annihilation operators only satisfy the canonical anticommutation relations of quantum field theory if they include a crucial sign (phase) factor that depends on the occupation numbers of all modes with lower indices. Definitions omitting this sign factor—such as those sometimes written in introductory texts or in Weinberg’s equations without explicit mention—do not, by themselves, satisfy the full set of anticommutation relations for multiple modes. The sign factor ensures antisymmetry under particle exchange, a fundamental property for fermions, and encodes the Pauli exclusion principle at the operator level.
Let’s unravel why this is the case, and how it plays out in both abstract mathematics and explicit matrix representations.
The Fock Space and Operator Action
To begin, recall that in the language of second quantization, states of a quantum field are described in Fock space—built from the vacuum by applying creation operators for each “mode” (which might be a momentum, spin, or spatial mode). As described on phys.libretexts.org, the Fock space basis states are written as |n₁, n₂, n₃, ...⟩, where n_j is the occupation number of mode j, either 0 or 1 for fermions, reflecting the Pauli exclusion principle.
The creation operator a_j^† adds a fermion to mode j if it is empty, while the annihilation operator a_j removes it if present. But for fermions, the action of these operators must also encode the antisymmetry of the wavefunction: swapping two identical fermions must introduce a minus sign.
This is where the algebra of the operators enters. The canonical anticommutation relations, as discussed on physicsforums.com and physics.stackexchange.com, are: { a_r, a_s^† } = a_r a_s^† + a_s^† a_r = δ_{rs} { a_r, a_s } = 0 { a_r^†, a_s^† } = 0
These rules ensure that two fermions cannot occupy the same mode, and that the overall wavefunction is antisymmetric under exchange.
Matrix Definitions and the Missing Sign
Now, how do we write these operators in matrix form? For a single mode, the creation and annihilation operators can be represented by 2x2 matrices. For one site, as shown in a discussion at physics.stackexchange.com, the annihilation operator f₀ is a matrix with a single off-diagonal 1, and the creation operator f₀^† is its transpose: f₀ = [ [0, 1], [0, 0] ] f₀^† = [ [0, 0], [1, 0] ]
These satisfy the single-mode anticommutation relation: { f₀, f₀^† } = 1.
However, for multiple modes, the situation is more complex. If you naively define each mode’s operator to act only on its own site (and as identity elsewhere), as in the Kronecker delta expressions from Weinberg’s text (see physics.stackexchange.com), you find a problem: for different modes j ≠ k, the operators commute, not anticommute. This violates the required algebra.
As stated in a post on physics.stackexchange.com, “Both terms give exactly the same non-zero result! … It seems these matrices commute rather than anticommute.” This flaw arises because, when acting on different sites, the operators do not “see” each other—there’s no mechanism for introducing the crucial minus sign when exchanging particles between modes.
The Solution: The Jordan-Wigner String
The resolution is the so-called Jordan-Wigner string, a phase factor that encodes the parity of the occupation numbers of all lower-indexed modes. The corrected operator action is (from physics.stackexchange.com and as summarized in multiple sources): a_k^† |n₁, n₂, ..., n_k, ...⟩ = (–1)^{Σ_{i<k} n_i} (1 – n_k) |n₁, n₂, ..., n_k+1, ...⟩ a_k |n₁, n₂, ..., n_k, ...⟩ = (–1)^{Σ_{i<k} n_i} n_k |n₁, n₂, ..., n_k–1, ...⟩
The phase factor (–1)^{Σ_{i<k} n_i} ensures that, when you swap the order of two operators acting on different modes, you pick up a minus sign, as required for fermions. This is not just a mathematical trick: it is essential for capturing the antisymmetric nature of the fermionic wavefunction.
As explained by users on physics.stackexchange.com, “thanks to the string η_α we have attached to them, [the operators] obey the right anticommutation relations.” The explicit form of η_α is a product over all lower-indexed modes, and, in matrix language, is often realized as a chain of Pauli Z matrices or their equivalents.
Concrete Examples and Verification
Let’s look at this in practice. Consider two modes, with basis states |0,0⟩, |1,0⟩, |0,1⟩, |1,1⟩. The creation operator for mode 0, with the proper string, acts as: a_0^† |0,0⟩ = |1,0⟩ a_0^† |0,1⟩ = |1,1⟩ a_0^† |1,0⟩ = 0 a_0^† |1,1⟩ = 0
The creation operator for mode 1 must include a minus sign if the first mode is occupied: a_1^† |0,0⟩ = |0,1⟩ a_1^† |1,0⟩ = –|1,1⟩
This minus sign is the manifestation of the phase factor, and without it, the anticommutation relations would fail.
Checking the algebra, as done in physics.stackexchange.com and physicsforums.com, you find: { a_0^†, a_1^† } = a_0^† a_1^† + a_1^† a_0^† = 0 { a_0, a_1^† } = 0 { a_j, a_j^† } = 1
These relations hold only if the phase factor is present.
Physical Meaning and the Pauli Principle
Why is this so important? The antisymmetry enforced by the sign factor directly encodes the Pauli exclusion principle. If you try to apply the creation operator twice to the same mode: a_j^† a_j^† |...⟩ = – a_j^† a_j^† |...⟩ ⇒ a_j^† a_j^† |...⟩ = 0
No state can have two or more quanta of the same species, as noted by discussants on physicsforums.com and studysmarter.co.uk, reflecting “the constraints imposed by the fermionic nature of the operators.”
This structure also ensures that physical states constructed by applying creation operators to the vacuum are antisymmetric under exchange of any two fermions, as required by the spin-statistics theorem and the definition of identical fermions (see phys.libretexts.org).
What Goes Wrong Without the Sign
If you omit the sign factor, as in the naive matrix definitions, the operators for different modes commute rather than anticommute. This is confirmed by explicit calculations in physics.stackexchange.com, where a user observes, “It seems these matrices commute rather than anticommute. Does this imply that the phase factor is simply omitted for brevity in Weinberg’s text, or is there a standard convention...?”
Indeed, in many texts, the sign is omitted in the initial definition for simplicity, with the understanding that the full Jordan-Wigner string must be included for the anticommutation relations to hold in systems with more than one mode.
Summary and Broader Perspective
So, to return to the original question: do the matrix definitions of fermionic creation and annihilation operators satisfy the anticommutation relations of quantum field theory? The answer is yes, but only if those definitions include the essential sign (phase) factor—often called the Jordan-Wigner string—reflecting the occupation numbers of all modes with lower indices. This factor is what allows the operators to properly anticommute across modes, ensuring the correct statistics and physical behavior of fermionic systems.
As stated in a concise phrase from physics.stackexchange.com, “thanks to the string η_α... [the operators] obey the right anticommutation relations,” and only with this structure do the matrix representations work as required.
This subtlety is not just a technical detail, but a deep reflection of the nature of identical particles, quantum statistics, and the structure of quantum field theory itself. It’s a beautiful example of how mathematical precision encodes profound physical principles, and why careful attention to definitions is essential in the foundations of quantum physics.