36.3 Continuum formalism

36.3 Continuum formalism

Concept links · terms present in this machine draft; source roles are unverified: vector space · dual space · complex inner product · Lie algebra · Hilbert space · symmetry group

The use of cutofs allows for a finite dimensional phase space and makes it possible to straightforwardly use the Bargmann-Fock quantization method. Such cutofs however introduce very significant problems, by making unavailable some of the continuum symmetries and mathematical structures that we would like to exploit. In particular, the use of an infrared cutof (periodic boundary conditions) makes the momentum space a discrete set of points, and this set of points will not have the same symmetries as the usual continuous momentum space (for instance in three dimensions it will not carry an action of the rotation group $SO(3),$ ). In our study of quantum field theory we would like to exploit the action of space-time symmetry groups on the state space of the theory, so need a formalism that preserves such symmetries. In this section we will outline such a formalism, without attempting a detailed rigorous version. One reason for this choice is that for the case of physically interesting interacting quantum field theories this continuum formalism is inadequate, since a rigorous definition will require first defining a finite, cutof version, then using renormalization group methods to analyze the very non-trivial continuum limit.

If we try and work directly with the infinite dimensional space of solutions of the free Schr¨odinger equation, for the three forms of the Fock space construction discussed in section 36.1.1 we find:

• The occupation number construction of Fock space is not available (since it requires a discrete basis).

• For the Bargmann-Fock holomorphic function state space and inner product on it, one needs to make sense of holomorphic functions on an infinite dimensional space, as well as the Gaussian measure on this space. See section 36.6 for references that discuss this.

• For the symmetric tensor product representation, one needs to make sense of symmetric tensor products of infinite dimensional Hilbert spaces ${\mathcal{H}}_1$ and the induced Hilbert space structure on such tensor products. We will adopt this point of view here, with details available in the references of section 36.6.

In the continuum normalization, an arbitrary solution to the free particle Schr¨odinger equation is given by

$$\psi (x,t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\alpha (p)e^{ipx}{e}^{-i\frac{p^2}{2m}t}dp\text{\hspace{1em}(36.11)}$$ $$\mathrm{At}t=0$$ $$\psi (x,0)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\alpha (p)e^{ipx}dp$$

which is the Fourier inversion formula, expressing a function $\psi (x,0)$ in terms of its Fourier transform, $\alpha (p)=\psi (x,0)(p)$ . We see that the functions $\alpha (p)$ parametrize initial data and ${\mathcal{H}}_1$ , the solution space of the free particle Schr¨odinger equation, can be identified with the space of such $\alpha$.

Using the notation $A(\alpha )$ to denote the element of ${\mathcal{H}}_1$ determined by initial data $\alpha ,$ in the Fock space description of multi-particle states as symmetric tensor products of ${\mathcal{H}}_1$ we have the following annihilation and creation operators (these were discussed in the finite dimensional case in section 26.4)

$$a^{†}(\alpha )P^{+}(A\left({\alpha }_1\right)\otimes \cdots \otimes A\left({\alpha }_n\right))=\sqrt{n+1}{P}^{+}(A(\alpha )\otimes A\left({\alpha }_1\right)\otimes \cdots \otimes A\left({\alpha }_n\right))\text{\hspace{1em}(36.12)}$$

$$\begin{array}{l}a(\alpha )P^{+}(A({\alpha }_1)\otimes \cdots \otimes A({\alpha }_n))=\\ \frac{1}{\sqrt{n}}{\sum }_{j=1}^n⟨\alpha ,{\alpha }_j⟩P^{+}(A({\alpha }_1)\otimes \cdots \otimes \widehat{A({\alpha }_j)}\otimes \cdots \otimes A({\alpha }_n))\end{array}\text{\hspace{1em}(36.13)}$$

(the $\widehat{A({\alpha }_j)}$ means omit that term in the tensor product, and $P^{+}$ is the symmetrization operator defined in section 9.6) satisfying the commutation relations

$$\begin{array}{rl}[a({\alpha }_1),a({\alpha }_2)]& =[a^{†}({\alpha }_1),a^{†}({\alpha }_2)]=0\\ [a({\alpha }_1),a^{†}({\alpha }_2)]& =⟨{\alpha }_1,{\alpha }_2⟩=\int \overline{{\alpha }_1(p)}{\alpha }_2(p)dp\end{array}\text{\hspace{1em}(36.14)}$$

Diferent choices for which space of functions $\alpha$ to take as ${\mathcal{H}}_1$ lead to diferent problems. Three possibilities are:

$$\bullet {\mathcal{H}}_1=L^2(\mathbf{R})$$

This choice allows for an isomorphism between ${\mathcal{H}}_1$ and its dual, using the inner product

$$⟨{\alpha }_1,{\alpha }_2⟩=\int \overline{{\alpha }_1(p)}{\alpha }_2(p)dp$$

As in the single-particle case the problem here is that position

$$\alpha (p)=\frac{1}{\sqrt{2\pi }}{e}^{ip{x}^{\prime }}$$

and momentum

$$\alpha (p)=\delta (p-p^{\prime })$$

eigenstates are not in ${\mathcal{H}}_1=L^2(\mathbf{R})$ . In addition, as in the single-particle case, there are domain issues to consider, since diferentiating by $p$ or multiplying by $p$ can take something in $L^2(\mathbf{R})$ to something not in $L^2(\mathbf{R})$

${\mathcal{H}}_1=\mathcal{S}(\mathbf{R})$

This choice, taking $\alpha$ to be in the well-behaved space of Schwartz functions, avoids the domain issues of $L^2(\mathbf{R})$ and the Hermitian inner product is well-defined. It however shares the problem with $L^2(\mathbf{R})$ of not including position or momentum eigenstates. In addition, the inner product no longer provides an isomorphism of ${\mathcal{H}}_1$ with its dual.

${\mathcal{H}}_1=S^{\prime }(\mathbf{R})$

This choice, allowing $\alpha$ to be distributional solutions, will solve domain issues, and includes position and momentum eigenstates. It however introduces a serious problem: the Hermitian inner product on functions does not extend to distributions. With this choice ${\mathcal{H}}_1$ is not an inner product space and neither are its symmetric tensor products.

To get a rigorous mathematical formalism, for some purposes it is possible to adopt the first choice, ${\mathcal{H}}_1=L^2(\mathbf{R})$ . With this choice the symmetric tensor product version of the Fock space can be given a Hilbert space structure, with operators $a^{†}(\alpha )$ and $a(\alpha )$ defined by equations 36.12 and 36.13 satisfying the Heisenberg commutation relations of equation 36.14. We will however want to consider operators quadratic in the $a(\alpha )$ and $a^{†}(\alpha )$ , and for these to be well defined we may need to use ${\mathcal{H}}_1=\mathcal{S}(\mathbf{R})$

If we ignore the problem with the inner product, and take ${\mathcal{H}}_1=S^{\prime }(\mathbf{R})$ , then in particular we can take $\alpha$ to be a delta-function, and when doing this will use the notation

$$a(p)=a(\delta (p^{\prime }-p)),a^{†}(p)=a^{†}(\delta (p^{\prime }-p))$$

and write

$$a(\alpha )=\int \overline{\alpha (p)}a(p)dp,a^{†}(\alpha )=\int \alpha (p)a^{†}(p)dp\text{\hspace{1em}(36.15)}$$

The choice of the conjugations here reflects that fact that $a^{†}(\alpha )$ is complex linear in $\alpha ,a(\alpha )$ complex antilinear.

While the operator $a(p)$ may be well-defined, the problem with the operator $a^{†}(p)$ is clear: it takes in particular the vacuum state |0⟩ to the non-normalizable state $|p⟩$ . We will, like most other authors, often write equations in terms of operators $a(p)$ and $a^{†}(p)$ , acting as if ${\mathcal{H}}_1={\mathcal{S}}^{\prime }(\mathbf{R})$ . For a legitimate interpretation though, such equations will always require an interpretation either

• using cutofs which make the values of $p$ discrete and of finite number, $p_j$ labeled by an index $j$ with a finite number of values, as in section 36.2. In this case the $a(p),a^{†}(p)$ are the $a(p_j),a^{†}(p_j)$ of that section.

• using equations 36.15 formally, with $a(\alpha )$ and $a^{†}(\alpha )$ the objects that are well-defined, for some specified class of functions $\alpha$, generally $\mathcal{S}(\mathbf{R})$ . In this case the $a(p),a^{†}(p)$ are often described as “operator-valued distributions”.

The non-zero commutators of $a(p),a^{†}(p)$ can be written as

$$[a(p),a^{†}(p^{\prime })]=\delta (p-p^{\prime })$$

a formula that should be interpreted as meaning either a continuum limit of

$$[a(p_j),a^{†}(p_k)]={\delta }_{jk}$$

or

$$\begin{array}{rl}\left[a({\alpha }_1),a^{†}({\alpha }_2)\right]& =\left[\int \overline{{\alpha }_1(p)}a(p)dp,\int {\alpha }_2(p^{\prime })a(p^{\prime })d{p}^{\prime }\right]\\ & =\int \int \overline{{\alpha }_1(p)}{\alpha }_2(p^{\prime })\delta (p-p^{\prime })dpd{p}^{\prime }\\ & =\int \overline{{\alpha }_1(p)}{\alpha }_2(p)dp=⟨{\alpha }_1,{\alpha }_2⟩\end{array}$$

for some class $(e.g.S(\mathbf{R})$ or $L^2(\mathbf{R}))$ of functions for which the inner product of ${\alpha }_1$ and ${\alpha }_2$ makes sense.

While we have defined here first the quantum theory in terms of a state space and operators, one could instead start by writing down a classical theory, with dual phase space ${\mathcal{H}}_1$ . This is already a complex vector space with Hermitian inner product, so we are in the situation described for the finite dimensional case in section 26.4. We need to apply Bargmann-Fock quantization in the manner described there, introducing a complex conjugate space ${\overline{\mathcal{H}}}_1$ , as well as a symplectic structure and indefinite Hermitian inner product on ${\mathcal{H}}_1\oplus {\overline{\mathcal{H}}}_1$ Restricted to ${\mathcal{H}}_1$ the Hermitian inner product will be the given one, and the symplectic structure will be its imaginary part.

If we denote by $A(\alpha )\in {\mathcal{H}}_1$ the solution of the Schr¨odinger equation with Fourier transform of initial data given by $\alpha$, and by $\overline{A}(\alpha )\in {\overline{\mathcal{H}}}_1$ the conjugate solution of the conjugate Schr¨odinger equation, the Poisson bracket relations are then

$$\{A({\alpha }_1),A({\alpha }_2)\}=\{\overline{A}({\alpha }_1),\overline{A}({\alpha }_2)\}=0,\{A({\alpha }_1),\overline{A}({\alpha }_2)\}=i⟨{\alpha }_1,{\alpha }_2⟩\text{\hspace{1em}(36.16)}$$

Quantization then takes

$$A(\alpha )\to -i{a}^{†}(\alpha ),\overline{A}(\alpha )\to -ia(\alpha )$$

where $a^{†}(\alpha )$ and $a(\alpha )$ are given by equations 36.12 and 36.13. This gives a representation of the Lie algebra relations 36.16 for an infinite dimensional Heisenberg Lie algebra.

As with annihilation and creation operators, adopting a notation that formally extends the state space to ${\mathcal{H}}_1=S^{\prime }(\mathbf{R})$ , we define

$$A(p)=A(\delta (p^{\prime }-p)),\overline{A}(p)=\overline{A}(\delta (p^{\prime }-p))$$ $$A(\alpha )=\int \alpha (p)A(p)dp,\overline{A}(\alpha )=\int \overline{\alpha (p)}\overline{A}(p)dp$$

with Poisson bracket relations written

$$\{A(p),A(p^{\prime })\}=\{\overline{A}(p),\overline{A}(p^{\prime }\}=0,\{A(p),\overline{A}(p^{\prime })\}=i\delta (p-p^{\prime })\text{\hspace{1em}(36.17)}$$

To get observables, we would like to define quadratic products of operators such as

$$\hat{N}={\int }_{-\infty }^{+\infty }a(p{)}^{†}a(p)dp$$

for the number operator,

$$\hat{P}={\int }_{-\infty }^{+\infty }pa(p{)}^{†}a(p)dp$$

for the momentum operator, and

$$\hat{H}={\int }_{-\infty }^{+\infty }\frac{p^2}{2m}a(p{)}^{†}a(p)dp$$

for the Hamiltonian operator. One way to make rigorous sense of these is as limits of the operators 36.8, 36.9 and 36.10. Another is as bilinear forms on $S^{\ast }({\mathcal{H}}_1)\times S^{\ast }({\mathcal{H}}_1)$ for ${\mathcal{H}}_1=\mathcal{S}(\mathbf{R})$ , sending pairs of states $|{\varphi }_1⟩,|{\varphi }_2⟩$ to, for instance, $⟨{\varphi }_1|\hat{N}|{\varphi }_2⟩$ (for details see [17], section 5.4.2).

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