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Insert the given closed-string mode expansion into the stress tensor and compare $T(z)={\sum }_n{L}_n{z}^{-n-2}$. Normal ordering gives

The inverse of the worldsheet Laplacian gives the free-field operator-product expansionHenceExtracting the Laurent modes by contour integration yields the string oscillator algebra

The vanishing of the worldsheet stress tensor becomes the Virasoro constraintsSince $L_0={\alpha }^{\prime }{k}^2/4+N$ and the bosonic-string normal-ordering constant is $a=1$, these zero-mode constraints are the target-space string mass-shell conditionThereforeThe positive-mode constraints further requireand null string states identify polarizations that differ by momentum-longitudinal terms. The symmetric trace-free, antisymmetric, and trace sectors of ${ϵ}_{\mu \nu }$ describe the graviton, Kalb–Ramond field, and dilaton, respectively.

Finally, the state–operator correspondence maps ${\alpha }_{-1}^{\mu }{\bar{\alpha }}_{-1}^{\nu }|k⟩$ toAdding its integrated massless closed-string vertex operator to the string nonlinear sigma model changes the background coupling byThus a Fourier mode $\delta G_{\mu \nu }(X)={ϵ}_{(\mu \nu )}{e}^{ik\cdot X}$ is precisely a linearized deformation of the target-space metric. Its transversality, mass-shell condition, and polarization gauge redundancy are the worldsheet statements that the deformation is marginal and is defined up to a linearized spacetime diffeomorphism.

In Feynman gauge, the Lagrangian differs by a total divergence fromIts Euler-Lagrange equation is $\square A^{\mu }=0$. Directly from the original gauge-fixed Lagrangian, the canonical momenta areequivalently ${\pi }^{\mu }=-{\partial }^0{A}^{\mu }$ after using the total-divergence form.

Using the mode expansions and oscillator commutator, the two nonzero cross terms combine towhere polarization completeness was used. The other equal-time field commutators vanish.

Substitution into the Hamiltonian and normal ordering giveEvery oscillator excitation carries positive energy $|\mathbf{p}|$, but the covariant state space has an indefinite inner product: time-like excitations have negative norm, and time-like and longitudinal polarizations are unphysical. Gupta-Bleuler quantization imposesand quotients by null states. The physical state space contains only the two transverse photon polarizations with positive norm, as required by gauge invariance.

Normal ordering defines a composite field by subtracting all singular self-contractions before bringing its constituent points together. For example,This makes $T$ and $\bar{T}$ well-defined local operators and is equivalent in oscillator language to placing creation operators to the left of annihilation operators.

Because normal ordering removes contractions within either exponential, Wick theorem sums all cross-contractions intoUnder $z\mapsto \lambda z$, this scales as ${\lambda }^{-k^2}$. Each operator has scaling dimension $\Delta =h+\tilde{h}=k^2/2$, so a two-point function of two such primaries must scale as ${\lambda }^{-2\Delta }={\lambda }^{-k^2}$. The two calculations agree.

For a restricted representation of an affine current algebra with $[J_m^a,J_n^b]=i{f}^{abc}{J}_{m+n}^c+km{\delta }^{ab}{\delta }_{m+n,0}$ and $f^{abc}{f}^{abd}=Q{\delta }^{cd}$, the displayed normal ordering construction is defined when $2k+Q\ne 0$. Moving the currents past each other gives $[L_m,J_n^a]=-n{J}_{m+n}^a$; the additional $Q$ in the denominator is the adjoint-Casimir reordering term. The resulting Virasoro algebra has central charge $c=k\dim \mathfrak{g}/(k+Q/2)$. On a compact unitary positive-energy module, $L_m^{†}=L_{-m}$.

Split a free scalar field into its annihilation and creation parts. For $x^0>y^0$, changing the product into normal ordering adds $[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$, the positive-frequency two-point function. For the opposite ordering interchange the two points. Their combination is the Feynman propagator, equal to the vacuum expectation of the time-ordered product. Iterating this identity gives the scalar Wick theorem.

Substituting the free complex scalar field modes into $i\int ({\varphi }^{†}\dot{\varphi }-{\dot{\varphi }}^{†}\varphi )d^{3}x$ gives the formally ordered expression $Q_{\mathrm{bare}}=\int d{\Pi }_p(a^{†}a-b{b}^{†})$. Its divergent vacuum constant is removed by normal ordering, equivalently by choosing $Q|0⟩=0$. The resulting complex scalar charge operator has $[Q,a^{†}]=a^{†}$ and $[Q,b^{†}]=-b^{†}$. Subtracting the constant changes neither its action on fields nor current conservation, but it is essential when identifying the vacuum charge and the charges of one-particle states.

Weyl ordering symmetrizes position and momentum factors in a classical polynomial before promoting them to an operator. Its phase-space symbol corresponds to midpoint time slicing of a phase-space path integral. In particular, the Weyl symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$, whereas its normal ordering symbol is $\bar{\psi }\psi$.