From Fock Space to Quantum Fields

Fock space organizes occupation states. A local quantum field additionally specifies how creation and annihilation operators are combined with spacetime modes. For a free real scalar, that construction produces a Hermitian field whose vacuum-to-one-particle matrix element is a positive-frequency Klein–Gordon wave amplitude. Its conserved wave norm in the field-theory normalization is exactly the particle state’s Hilbert norm. This is one precise sense in which one-particle wave mechanics survives inside a quantum field theory.

Required background. Fock Space and Occupation Number provides the state space; Relativistic Phase Space fixes the measure; Klein–Gordon Inner Product normalizes the extracted amplitude.

Helpful background. From Harmonic Oscillators to Fields checks mode counting; Locality and Causality Warnings gives the spacelike commutator calculation used below.

The free scalar field in a covariant mode convention

Use $\hbar=c=1$, metric $(+---)$, and a real scalar of mass $m>0$. All momentum labels below have $p^0=E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}>0$. Adopt the package

$$d\Pi_p=\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}},$$ $$[a(\mathbf p),a^\dagger(\mathbf q)] =2E_{\mathbf p}(2\pi)^3\delta^3(\mathbf p-\mathbf q),$$

with the other ladder commutators zero and $a(\mathbf p)|0\rangle=0$. Then $|\mathbf p\rangle=a^\dagger(\mathbf p)|0\rangle$ has the covariant momentum-state normalization. These are operator-valued distributions: mode equations are used on suitable wave packets, and local fields are smeared when defining operators on a common domain.

The oscillator construction gives

$$\widehat\phi(x) =\int d\Pi_p\left[ a(\mathbf p)e^{-ip\cdot x} +a^\dagger(\mathbf p)e^{ip\cdot x}\right].$$

It is Hermitian and obeys $(\Box+m^2)\widehat\phi=0$ distributionally. The real scalar has one neutral species; the two terms do not introduce a second antiparticle species. Their operator roles are annihilation and creation.

In this convention the normal-ordered free operators are

$$\begin{aligned} H_0&=\int d\Pi_p\,E_{\mathbf p}a^\dagger(\mathbf p)a(\mathbf p),\\ \mathbf P&=\int d\Pi_p\,\mathbf p\,a^\dagger(\mathbf p)a(\mathbf p),\\ N&=\int d\Pi_p\,a^\dagger(\mathbf p)a(\mathbf p). \end{aligned}$$

The free reference vacuum energy has been treated separately. The detailed oscillator and normal-ordering construction is in the Reference field bridge. Using the noncovariant $N$-convention operators in this expansion without rescaling them would change its factors: $a_{\rm cov}(\mathbf p)=\sqrt{2E_{\mathbf p}}\, a_N(\mathbf p)$ when $[a_N,a_N^\dagger]=(2\pi)^3\delta^3$. The normalization ledger reserves its $\delta$ label for a bracket with a bare $\delta^3$ and no $(2\pi)^3$.

A field matrix element is a wave amplitude

Let $f$ be a smooth rapidly decreasing wave-packet coefficient, and define

$$|f\rangle=\int d\Pi_p\,f(\mathbf p) a^\dagger(\mathbf p)|0\rangle.$$

One use of the ladder commutator gives

$$\langle f|f\rangle=\int d\Pi_p\,|f(\mathbf p)|^2.$$

The same contraction in a vacuum-to-particle matrix element yields

$$F_f(x)=\langle0|\widehat\phi(x)|f\rangle =\int d\Pi_p\,f(\mathbf p)e^{-ip\cdot x}.$$

Only the annihilation part contributes. Consequently $F_f$ is a positive-frequency Klein–Gordon solution. The field itself contains both frequency signs; the choice of bra and ket selected one sign in this particular matrix element.

Use the field-theory normalization of the conserved Klein–Gordon form,

$$(F_f,F_f)_{\rm KG,QFT} =i\int d^3x\,F_f^* \overleftrightarrow{\partial_t}F_f.$$

In natural units this is $2m$ times the one-particle form used as the main convention on Klein–Gordon Inner Product. That owner’s final normalization comparison distinguishes the two forms. The unrescaled field matrix element here uses the field-theory convention throughout.

The spatial Fourier integral supplies $(2\pi)^3\delta^3(\mathbf p-\mathbf q)$ and the two time derivatives supply $E_{\mathbf p}+E_{\mathbf q}$. With the covariant measures this gives

$$(F_f,F_f)_{\rm KG,QFT} =\int d\Pi_p\,|f(\mathbf p)|^2 =\langle f|f\rangle.$$

Thus a normalized one-particle state has a normalized positive-frequency wave amplitude in the field-theory KG norm. This does not promote $|F_f(x)|^2$, or the pointwise signed KG density, to a general local relativistic position probability. The localization limitations remain relevant.

For two packets the same calculation gives $(F_f,F_g)_{\rm KG,QFT}=\langle f|g\rangle$. The map preserves their interference and inner product; it is not merely an analogy between similar differential equations.

The field can change number while free evolution preserves it

The ladder algebra implies $[N,a]=-a$ and $[N,a^\dagger]=a^\dagger$. Writing $\widehat\phi=\widehat\phi^{(+)} +\widehat\phi^{(-)}$, where the superscripts denote the annihilation and creation frequency parts, gives

$$[N,\widehat\phi] =-\widehat\phi^{(+)}+\widehat\phi^{(-)}.$$

A field insertion therefore connects different number sectors. But $[H_0,N]=0$: free time evolution preserves them. A number-changing observable or insertion is not the same as a number-changing Hamiltonian.

For example, $\langle f|\widehat\phi(x)|f\rangle=0$ in a one-particle state, since the operator maps it into zero- and two-particle sectors. Its vacuum-to-particle matrix element $F_f(x)$ can nevertheless be nonzero. An interaction such as a regulated local polynomial in the field can connect sectors dynamically; its definition and renormalization are additional work.

Locality fixes more than the list of oscillators

In this free construction the field commutator is

$$[\widehat\phi(x),\widehat\phi(y)] =C(x-y)I,$$ $$C(z)=\int d\Pi_p \left(e^{-ip\cdot z}-e^{ip\cdot z}\right).$$

The spacelike cancellation and the distinction from nonzero Wightman correlations were proved in Locality and Causality Warnings. The mode algebra here supplies the quantum-field input to that calculation. At equal time, the same normalization gives $[\widehat\phi(t,\mathbf x), \partial_t\widehat\phi(t,\mathbf y)] =i\delta^3(\mathbf x-\mathbf y)$.

By contrast, keeping only the annihilation part gives

$$[\widehat\phi^{(+)}(x), \widehat\phi^{(+)}(y)^\dagger] =\int d\Pi_p\,e^{-ip\cdot(x-y)}.$$

This generally does not vanish spacelike. It uses the same Fock operators but does not have the local real field’s algebra. Fock-space organization alone therefore does not choose the local field: the mode weights, both frequency parts, and transformation law must also be specified.

Smearing produces a normalizable packet

For a smooth rapidly decreasing spacetime test function $g$, define

$$\widehat\phi(g)=\int d^4x\,g(x)\widehat\phi(x), \qquad \widetilde g(p)=\int d^4x\,g(x)e^{ip\cdot x}.$$

Acting on the vacuum gives

$$\widehat\phi(g)|0\rangle =\int d\Pi_p\,\widetilde g(p) a^\dagger(\mathbf p)|0\rangle.$$

Only the positive mass-shell restriction of $\widetilde g$ enters this state, and its norm is $\int d\Pi_p|\widetilde g(p)|^2$. Different test functions with the same restriction therefore create the same free one-particle packet.

A formal point insertion instead has norm squared proportional to $\int d\Pi_p$, which diverges in $3+1$ dimensions. With a large momentum cutoff $\Lambda$, its leading growth is $\Lambda^2/(8\pi^2)$. The object $\widehat\phi(x)|0\rangle$ is not a normalized particle localized at a mathematical point. Smearing makes the field operation meaningful, while the resulting state’s localization still obeys the relativistic qualifications above.

This exact construction is for a free massive field and its chosen Fock representation. Interacting fields, physical vacuum selection, asymptotic particles, and infrared limits require further structure. The fact that a free particle amplitude is a field matrix element does not prove that every interacting state has a finite-particle free-field description.

Exercises

Changing normalization conventions. If $|f\rangle=\int d^3p/(2\pi)^3\, h(\mathbf p)a_N^\dagger(\mathbf p)|0\rangle$, find $h$ in terms of the covariant coefficient $f$.

Solution

Substitute $a_{\rm cov}^\dagger=\sqrt{2E_{\mathbf p}}\, a_N^\dagger$ into the covariant state integral. Then $h=f/\sqrt{2E_{\mathbf p}}$, and $\int d^3p/(2\pi)^3|h|^2=\int d\Pi_p|f|^2$.

The free equation as a smearing redundancy. For smooth rapidly decreasing $h(x)$, show that test functions $g$ and $g+(\Box+m^2)h$ create the same state from the vacuum.

Solution

Integration by parts gives the Fourier difference $(m^2-p^2)\widetilde h(p)$. It vanishes on the mass shell that enters the state. In the free theory this also expresses the distributional field equation.

One-particle amplitude versus one-point mean. Can a normalized one-particle packet have $\langle f|\widehat\phi(x)|f\rangle=0$ for all $x$ while $F_f(x)$ is nonzero?

Solution

Yes. The diagonal mean vanishes by particle-number orthogonality, while the vacuum-to-one-particle matrix element selects the allowed annihilation transition. They are different matrix elements of the same operator.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press (2014). doi:10.1017/9781139540940. Free-field normalization and particle matrix elements.
• Tong, David. Quantum Field Theory. University of Cambridge lecture notes (2006), sections 2.2–2.7. Free fields. Scalar field operators, relativistic states, locality, and propagators.