Geometrical Formulation

Previously we introduced the multi-dimensional Virasoro algebra in a Fourier basis on the \(d\)-dimensional torus, and showed that trying to build representations in the same way as in one dimension fails. Today I will describe a more geometrical formulation which will make the connection to QJT explicit.

Infinitesimal diffeomorphisms are generated by vector fields, which are locally of the form
\[
\xi = \xi^\mu\partial_\mu 
= \sum_{\mu=0}^{d-1} \xi^\mu(x) {\partial\over\partial x^\mu},
\]
where again we use Einstein's summation convention: two indices, one up and one down, are implicitly summed over. The bracket between two vector fields is
\[
[\xi, \eta] = \xi^\mu \partial_\mu \eta^\nu \partial_\nu -
\eta^\nu \partial_\nu \xi^\mu \partial_\mu.
\]
The diffeomorphism algebra, or algebra of vector fields, is the Lie algebra generated by the Lie derivatives \({\cal L}_\xi\):
\[
[{\cal L}_\xi, {\cal L}_\eta] = {\cal L}_{[\xi,\eta]}.
\]
In this formalism, the multi-dimensional Virasoro algebra takes the form in the blog banner.
\[\begin{align}
[{\cal L}_\xi, {\cal L}_\eta] &= {\cal L}_{[\xi,\eta]}
 + {1\over{2\pi i}}\int dt\ \dot q^\rho(t)
 \Big( c_1\ \partial_\rho\partial_\nu\xi^\mu(q(t))\ \partial_\mu \eta^\nu(q(t))\ +\\
& \qquad\qquad\qquad +\ c_2\ \partial_\rho \partial_\mu \xi^\mu(q(t))\ \partial_\nu \eta^\nu(q(t)) \Big),\\
[{\cal L}_\xi, q^\mu(t)] &= \xi^\mu(q(t)), \\
[q^\mu(t), q^\nu(t')] &= 0. 
\end{align}\] 
This is strictly speaking not a Lie algebra, because \(q\) appears non-linearly in the right-hand side. But we can extend these brackets to arbitrary functionals of \(q\), and then they define a proper Lie algebra.

To verify that this is a Lie algebra (anti-symmetry and Jacobi identities), we integrate by parts and throw away the boundary terms. Hence we must also add the condition that the integral of a total derivative vanishes.
\[
\int dt {dF(t)\over dt} \equiv 0.
\]
This is the case if the integral runs over a circle, or if we only consider vector fields which decrease sufficiently fast when \(t \to \pm\infty\). 

To connect this formulation with the previous one, we note that a basis for the vector fields on the \(d\)-torus is given by the plane waves \(i\exp(im\cdot x) \partial_\mu\). Define
\[\begin{align}
L_\mu(m) &= {\cal L}(i\exp(im\cdot x) \partial_\mu) \\
S^\mu(m) &= {1\over{2\pi}}\int dt\ \dot q^\mu(t) \exp(im\cdot q(t)) 
\end{align}\] 
These operators indeed satisfy the multi-dimensional Virasoro algebra in the Fourier basis. In particular, the vanishing of total derivatives leads to the condition
\[
m_\mu S^\mu = 
{1\over{2\pi i}} \int dt\ {d\over dt}\Big(\exp(im\cdot q(t)) \Big) 
\equiv 0.
\]
We used the same letter \(q\) here as we did for the expansion point in the Taylor series. This is not a coincidence, and we will show later that they are the same. Hence the observer's position appears already in the very definition of the multi-dimensional Virasoro algebra, which explains why we need QJT to construct representations.

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Divergencies

Representations of the algebra of diffeomorphisms on the circle is essentially QFT on the circle. It would be natural to think that the same is true in higher dimensions, but this is not the case. If we try to repeat the recipe used in one dimension, we formally arrive at an infinite central charge, due to an unrestricted sum over transverse directions. This is of course a nonsense result, and it stopped physicists, myself included, from going further. Normal ordering only works if we start from a classical representation acting on finitely many functions of a single variable, because then there are no transverse directions that cause infinities. In later posts I will show how to do that within QJT.

First we review the construction of representations of the Virasoro algebra in one dimension. Let us start with the Heisenberg algebra:
\[\begin{align}
[\phi_m, \pi_n] &= i\delta_{m+n}, \\
[\phi_m, \phi_n] &= [\pi_m, \pi_n] = 0,
\end{align}\]
where \(m\) and \(n\) are integers. The algebra of diffeomorphisms on the circle can be embedded in this algebra in the following way.
\[
L_m = i \sum_{n=-\infty}^\infty (m-n) \phi_{m-n} \pi_n
\]
The brackets with the elements of the Heisenberg algebra read
\[\begin{align}
[L_m, \phi_n] &= (m+n) \phi_{m+n}, \\
[L_m, \pi_n] &= n \pi_{m+n},
\end{align}\]
which are the transformation laws of a scalar field and a scalar density, respectively.

Embedding the diffeomorphism algebra into the Heisenberg algebra means that a representation of the latter gives rise to a representation of the former. Introduce a vacuum \(|0>\) which annihilates all negative modes.
\[
\phi_{-m} |0> = \pi_{-m} |0> = 0,
\]
for all \(m > 0\).  This does not quite give us a representation of the diffeomorphism algebra, because the vacuum energy is infinite.
\[
L_0 |0> = -i\sum_{n=1}^\infty n \phi_{-n} \pi_n |0>
= \sum_{n=1}^\infty n |0>.
\]
To avoid this problem we redefine \(L_0\) so that \(L_0 |0> = 0\). 
\[
L_0 = i \sum_{n=1}^\infty n 
  \Big( \phi_{n} \pi_{-n} - \pi_{n} \phi_{-n}\Big)
\]
This is called normal ordering, because it is equivalent to moving the infinitely many negative modes to the right. However, this changes the algebra that the generators \(L_m\) satisfy, which can be seen by calculating the bracket
\[
[L_m, L_{-m}] |0> = -\sum_{n=1}^{m-1} n(m-n)|0> = -{(m^3-m)\over6}|0>
\]
Comparing with the definition of the Virasoro algebra in the previous post we see that normal ordering leads to a central charge \(c = 2\). The central charge is positive because the construction was made with bosonic fields. Repeating the analysis with fermionic fields would lead to  \(c = -2\) instead.

Now let us try to repeat the analysis in two dimensions. The indices now become 2D integer vectors \(m = (m_0, m_1)\), and the diffeomorphism generators can be embedded in a Heisenberg algebra like this:
\[
L_\mu(m_0,m_1) = i \sum_{n_0=-\infty}^\infty 
\sum_{n_1=-\infty}^\infty (m_\mu-n_\mu) \phi(m_0-n_0,m_1-n_1) \pi(n_0,n_1)
\]
To build a representation we must introduce a vacuum which is annihilated by half the modes, e.g. those with \(m_0 < 0\). Again this leads to an infinite vacuum energy, which can be removed by moving modes with \(m_0 < 0\) to the right. But this kind of normal ordering goes wrong, because we have to reorder infinitely many terms. E.g., when we compute \([L_0(m_0,m_1), L_0(-m_0,-m_1)] |0>\) we formally get the central charge
\[
\sum_{n_0=1}^{m_0} \sum_{n_1=-\infty}^\infty n_0(m_0-n_0) = K {(m_0^3-m_0)\over6},
\]
where
\[
K = \sum_{n_1=-\infty}^\infty 1 = \infty.
\]
So the would-be central charge is infinite, due to the unrestricted sum over transverse modes. An infinite central charge is of course a sign that something has gone very wrong. Also, the diffeomorphism algebra in \(d > 1\) dimension does not have a central extension, because it would not be compatible with the Jacobi identities.

The conclusion is that representations of the Virasoro algebra in \(d > 1\) dimensions can not be built in the same way as we do when \(d=1\). This is because the natural arena is not QFT but QJT.

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Multi-dimensional Virasoro Algebra

Yesterday we discussed the Virasoro algebra, the central extension of the algebra of diffeomorphisms on the circle. A natural question is if this extension has a generalization to higher dimensions, e.g. to the diffeomorphism algebra on the \(d\)-dimensional torus where \(d > 1\). The answer is yes, but it is not straightforward, for two reasons:

• The diffeomorphism algebra in \(d\) dimensions has no central extension. The Virasoro extensions are not central except when \(d=1\).
• There are two independent extensions. This happens because the diffeomorphism group in \(d\) dimensions is closely related to the general linear group \(GL(d)\). There are two extensions because \(GL(d) = SL(d) \times GL(1)\) is a direct product. Only when \(d=1\) one extension vanishes because \(SL(1)\) is the trivial group consisting of a single element.

Yesterday we discussed the ordinary Virasoro algebra. The generators are \(L_m = -i\exp(imx) {d\over dx}\) and the brackets are
\[
[L_m, L_n] = (n-m) L_{m+n} - {c\over 12} m^3 \delta_{m+n}
\] 
where \(\delta_m\) is the Kronecker delta. Note that we have eliminated the linear term in the extension by a redefinition of \(L_0\).

Here comes the trick. We rewrite the ordinary Virasoro algebra as follows:
\[\begin{align}
[L_m, L_n] &= (n-m) L_{m+n} + c m^2 n S_{m+n}, \\
[L_m, S_n] &= (n+m) S_{m+n}, \\
[S_m, S_n] &= 0, \\
m S_m &\equiv 0.
\end{align}\] 
To see that this is the same Lie algebra as the previous one, we notice that the last condition implies that \(S_m\) is only nonzero when \(m=0\). Hence \(S_m = {1\over 12}\delta_m\), up to a factor. We then have \([L_m, S_n] = 0\), so the extension does indeed commute with everything. Finally the extension in the first line reduces to
\[
- {c\over12} m^3 \delta_{m+n}.
\]
The advantage of the second formulation is that it can be generalized to multiple dimensions. A basis for the vector fields in \(d\) dimensions are \(L_\mu(m) = -i\exp(im\cdot x) \partial_\mu\), where \(m\) and \(x\) are now \(d\)-dimensional vectors with components \(m_\mu\) and \(x^\mu\), and \(\partial_\mu = \partial/\partial x^\mu\). We use Einstein's summation convention, so repeated indices, one up and one down, are implicitly summed over. E.g.
\[ 
m \cdot x = m_\mu x^\mu = \sum_{\mu = 0}^{d-1} m_\mu x^\mu.
\]
The diffeomorphism algebra on the \(d\)-dimensional torus admits the following Virasoro-like extensions:
\[\begin{align}
[L_\mu(m), L_\nu(n)] &= n_\mu L_\nu(m+n) - m_\nu L_\mu(m+n) \\
&\qquad +\ (c_1 m_\nu n_\mu + c_2 m_\mu n_\nu)\ m_\rho S^\rho(m+n), \\
[L_\mu(m), S^\nu(n)] &= n_\mu S^\nu(m+n) + \delta^\nu_\mu m_\rho S^\rho(m+n), \\
[S^\mu(m), S^\nu(n)] &= 0, \\
m_\mu S^\mu(m) &\equiv 0.
\end{align}\] 
To verify that this is indeed a Lie algebra, we must check that the brackets are anti-symmetric and satisfy the Jacobi identity. We must also confirm that the last condition is preserved, and indeed it is:
\[
[L_\mu(m), n_\nu S^\nu(n)] = n_\mu (m_\nu + n_\nu) S^\nu(m+n) \equiv 0.
\]
Finally, we must show that these extensions are non-trivial, i.e. that they can not be absorbed into a redefinition of some generators. To this end, consider the subalgebra generated by the operators
\[
L_r = \alpha^\mu L_\mu(r\beta), \qquad
S_r = \beta_\nu S^\nu(r\beta),
\]
where \(\alpha^\mu\) and \(\beta_\nu\) are two constant vectors which satisfy \(\alpha^\mu\beta_\mu = 1\) and \(r\) is an integer. Then the \(L_r\) satisfies the ordinary Virasoro algebra with central charge
\[
c = c_1 + c_2.
\]
Since the restrictions of the two extensions of the full multi-dimensional Virasoro algebra are non-trivial, so are the extensions themselves.

The extension proportional to \(c_1\) was discovered by Rao and Moody [Comm. Math. Phys. 159 (1994) 239-264] and the one proportional to \(c_2\) by myself [J. Phys. A. 25 (1992) 1177–1184].

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Virasoro Algebra

The Virasoro algebra is a Lie algebra that appears in several places in theoretical physics, e.g. in string theory and 2D statistical physics. It was discovered independently by physicists and mathematicians in 1968.

The infinite conformal group in 2D is the group of analytic functions in the complex plane. A basis for the corresponding Lie algebra is given by
\[
L_m = z^{m+1} {d\over dz},
\]
where \(z\) is a point in the complex plane and \(m\) is an integer.
The algebra of diffeomorphisms on the circle has a Fourier basis
\[
L_m = -i \exp(imx) {d\over dx}.
\]
These two algebras are in fact the same (they are isomorphic), because the generators \(L_m\) have the same bracket. 
\[
[L_m, L_n] = (n-m) L_{m+n}.
\]
Recall that \([A,B] = AB - BA\). My sign convention is appropriate for lowest-weight representations. In the physics literature the opposite convention is more common.

This algebra of vector fields is not so interesting in itself. What is interesting is its central extension, the Virasoro algebra. So what is a central extension? An extension is a modification of the bracket that is still a Lie algebra. The extension is non-trivial if it cannot be eliminated with a change of basis, and it is central if its bracket with everything in sight vanishes. The Virasoro algebra is defined by
\[
[L_m, L_n] = (n-m) L_{m+n} - {c\over 12}(m^3 - m)\delta_{m+n},
\]
where \(\delta_m\) is the Kronecker delta, i.e. \(\delta_0 = 1\) and \(\delta_m = 0\) for all \(m \neq 0\). It is straightforward to verify that this bracket satisfies the conditions of a Lie algebra, i.e. anti-symmetry and the Jacobi identify still hold. The linear term in the extension is trivial, because it is eliminated if we redefine
\[
L_0 \Rightarrow L_0 + {c\over 24}.
\]
The cubic term can not be eliminated by a change of basis, so it is a non-trivial extension.

Tomorrow I will explain how to generalize the Virasoro algebra to higher dimensions. The infinite conformal group can not be generalized because it is specific to 2D, but the diffeomorphism algebra on the circle can. But the generalization is not straightforward; it took me several years and I still only got a partial answer at the time.

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Lie Groups and Algebras

Now it is time for some mathematical background.

A group is a set of objects that can be multiplied and have an inverse. 

A Lie group is a group that depends on continuous parameters. The name comes from Norwegian mathematician Sophus Lie and is pronounced Lee.

The simplest Lie group is the special orthogonal group SO(2), which describes rotations in the 2D plane. It is isomorphic (essentially the same as) the group U(1) of 1x1 unitary matrices, which can be identified with complex numbers of unit length. A rotation by an angle \(\varphi\) in the plane can be identified with multiplication by the complex number \(\exp(i\varphi)\).

A Lie group that is umbiquous in 3D software is the special orthogonal group SO(3), which describes rotations in 3D space. This is the group of real 3x3 matrices with unit determinant, where the inverse is the transposed matrix.

The special unitary group SU(2) of complex 2x2 matrices is the double cover of SO(3). This means that the groups are locally the same, but two separate SU(2) matrices correspond to the same SO(3) matrix. Quaternions are SU(2) matrices, and hence there are two different quaternions that correspond to a given rotation in 3D space.

A Lie algebra is the infinitesimal form of a Lie group. The basic operation in a Lie algebra is the commutator (or Lie bracket), given by
\[
[A, B] = AB - BA
\]
From the definition and associativity of matrix multiplication (\((AB)C = A(BC) = ABC\)) follow two conditions that any Lie algebra must obey.

Anti-symmetry: 
\[
[B, A] = -[A, B].
\]
Jacobi identify:
\[
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0.
\]

The infinitesimal form of the group SU(2) is the Lie algebra su(2), with three elements \(\sigma_i, i = 1,2,3\) (the Pauli matrices) that satisfy the brackets
\[
[\sigma_i, \sigma_j] = \sum_{k=1}^3 \epsilon_{ijk}\ \sigma_k,
\]
where \(\epsilon_{ijk}\) equals \(1\) if \(ijk\) is a positive permutation of \(123\), \(-1\) if it is a negative permutation, and \(0\) if two indices are equal. It is straightforward to verify that this bracket obeys anti-symmetry and the Jacobi identities.

A group which will play a prominent role in the future is the diffeomorphism group, i.e. the group of invertible maps of \(d\)-dimensional space to itself. Let \(F\) and \(G\) be two such maps. They form a group, with multiplication given by composition
\[
(F \circ G)(x) = F(G(x))
\]
The associated Lie algebra is the algebra of vector fields, which I informally call the diffeomorphism algebra. A vector field is locally of the form \(\xi = \xi^\mu(x) \partial_\mu\), where \(\partial_\mu = \partial/\partial x^\mu\) is the partial derivative, and I use Einstein's summation convention: repeated indices, one up and one down, are implicitly summed over.

The diffeomorphism algebra is defined by the brackets
\[
[\xi, \eta] = \xi^\mu \partial_\mu \eta^\nu \partial_\nu 
- \eta^\nu \partial_\nu \xi^\mu \partial_\mu.
\]
It is straightforward to verify that this is a Lie algebra (anti-symmetry and Jacobi identifies).

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Second Week

Horizontal fuzziness and Quantum Jet Theory (QJT) may seem to be ad hoc ideas coming out of nowhere. However, they are an inevitable consequence of the mathematics of the diffeomorphism group in multiple dimensions. During the first week I tried to keep the math down to a minimum, but now we have reached a point when some mathematical concepts and formulas are unavoidable. This week I plan to discuss the following topics:

• Lie Groups and Algebras
• Virasoro Algebra
• Multi-dimensional Virasoro Algebra
• Divergencies
• Geometrical Formulation
• Off-shell Representations
• Multi-dimensional Virasoro Algebra and Jets

Before digging too much into the math, I would like to give a birds-eye view of results that I want to present this week. The discussion will center around something called Lie algebras (I will explain what that is tomorrow) and their representations. These are closely related to quantum physics. Essentially, the representation theory of

• finite-dimensional Lie algebras is quantum mechanics, i.e. Quantum Field Theory (QFT) in zero dimensions.
• infinite-dimensional Lie algebras living on the circle, e.g. the Virasoro or affine Kac-Moody algebras, is QFT on the circle.
• infinite-dimensional Lie algebras living on higher-dimensional spaces is not QFT in the same dimension. Instead it is QJT.

The 1980s was a time of great change in theoretical physics, with conformal field theory transforming both string theory and statistical physics. Many physicists became interested in the Virasoro algebra, and quite a few tried to generalize it to diffeomorphisms in higher dimensions, using the same methods that worked on the circle. We all failed. At this point in my life I was running out of funding, and instead of moving and trying to pursue a very uncertain academic career, it was much more attractive to get a stable income and start a family.

A few years later I came across a paper by Rao and Moody [Comm. Math. Phys. 159: 239-264 (1994)] who claimed to have generalized the Virasoro algebra and constructed representations thereof, something that I had utterly failed to do myself. In fact, Rao had already sent me a preprint of their paper a few years before, but at that time I didn't grasp what they had done. This time I realized that their work was important, but it still took years to understand their formalism and its geometrical meaning. It turns out that their construction is equivalent to a limited form of QJT, which only involves the base point. In two articles I then generalized their result to full QJT, first to zero-jets [Comm. Math. Phys. 201, 461-470 (1999)] and then to arbitrary p-jets [Comm. Math. Phys. 214 469-491 (2000)]. Unfortunately the articles are behind a paywall, but preprints are available at the arXiv [physics/9705040], [math-ph/9810003].

It is now clear why the early attempts to apply QFT methods to the diffeomorphism algebra in multiple dimensions were doomed to failed. We were trying to apply QFT to a problem that requires QJT. The extension that generalizes the Virasoro algebra to higher dimensions is a functional of the observer's trajectory, so ignoring horizontal fuzziness is not possible. Except in one dimension, where the only possible trajectory is the circle itself.

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Partial and Complete Observables

Today I will repeat the argument using Rovelli's language of partial and complete variables [gr-qc/0110035], which I found useful to organize my thoughts.

Rovelli distinguishes between partial observables, which can be measured but not predicted, and complete observables, which can be predicted once we know the initial conditions. In a simple mechanical system there are two partial observables:

• \(\phi\): Result, measured by a detector.
• \(t\): Time, measured by a clock.

The partial observables can be measured but not predicted. What can be predicted is the complete observable \(\phi(t)\), the reading of the detector at time \(t\). In the quantum case predictions are statistical, so complete observables are fuzzy.

Now let us turn to field theory. There are now three partial observables:

• \(\phi\): Result, measured by a detector.
• \(t\): Time, measured by a clock.
• \(x\) or \(q\): Position, measured e.g. by a ruler or a GPS receiver.

In QFT we construct the complete observable \(\phi(x,t)\), the reading of the detector at time \(t\) and location \(x\), from these three partial observables. 

Here is the problem with QFT. The experiment takes place inside a detector, and we can in fact predict its future location. The detector is part of the physical world and has dynamics of its own. Instead we can construct two different complete observables from the same three partial observables:

• \(q(t)\): The detector's location at time \(t\).
• \(\phi(t)\): The reading of the detector at time \(t\), at the detector's location.

We can think of \(\phi(t)\) as \(\phi(q(t),t)\), the value of the field at the detector's location. Since complete observables are fuzzy, we now have two types of fuzziness:  the horizontal fuzziness of \(q(t)\) and the vertical fuzziness of \(\phi(t)\). In QFT we only have vertical fuzziness.

The obvious objection is that all information about the field outside the detector's worldline is lost. However, we can add more partial observables which can also be measured inside the detector:

• \(\phi_m\): The \(m\):th derivative of the field, also measured inside the detector.

From the complete observables \(\phi_m(t)\) we can construct the generating function
\[
\phi(x,t) = \sum_m {1\over m!} \phi_m(t) (x-q(t))^m,
\]
which can be viewed as the Taylor series expansion of the field \(\phi(x,t)\). To the extent an infinite Taylor series is equivalent to the field itself, we recover the field that we started from.

Hence we again arrive at Quantum Jet Theory.

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