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 \(L_m = -i\exp(imx)\) 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:
\[
[L_m, n_\nu S^\nu(n)] = n_\mu (m_\nu + n_\nu) S^\nu(m+n) \equiv 0.
\]
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
\[
c = 12(c_1 + c_2).
\]
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 QJT

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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Quantum Jet Theory

In the previous posts I argued that what is missing from Quantum Field Theory (QFT) is horizonal fuzziness. Today it is finally time to discuss how this can be implemented. Obviously, horizontal fuzziness requires that the theory has horizontal observables. QFT only has vertical observables, the values of the field at given points in space and time, and hence we can not formulate horizontal fuzziness within the framework of QFT. But there is a simple way to introduce a horizontal observable: expand the field in a Taylor series. A Taylor series has both vertical observables (the Taylor coefficients) and a horizontal observable (the base point a.k.a. the detector's position), and hence it can express both vertical and horizontal fuzziness. This is the idea behind Quantum Jet Theory (QJT). The name comes from the notion of jets in mathematics; a \(p\)-jet is locally the same as a Taylor series truncated at order \(p\). There might be other ways to include the detector's trajectory in a theory, but QJT is what naturally arises from the representation theory of the multi-dimensional Virasoro algebra that I will discuss next week.

QFT is quantum physics applied to fields, i.e. systems with infinitely many degrees of freedom. A field \(\phi(x,t)\) represents the result of an experiment \(\phi\) done at the spacetime point \((x,t)\). The result \(\phi\) is fuzzy, but the location \(x\) is known precisely at all times. Hence QFT has vertical fuzziness but no horizontal fuzziness. In QJT we expand the field in a Taylor series,
\[
\phi(x,t) = \sum_m {1\over m!} \phi_m(t) (x-q(t))^m.
\]
This formulation may seem to be limited to one dimension, but with multi-index notation it makes sense in higher dimensions as well.

The Taylor coefficients \(\phi_m\) are the vertical observables. They describe the result of the experiment inside the detector. The corresponding momenta \(\pi^m\) describe the experiment's rate of change. Here we put the index upstairs, because the momenta belong to the dual space. The product of the uncertainties obey
\[
\Delta \phi_m \cdot \Delta \pi^n \geq {\hbar\over2} \delta^n_m
\]
where \(\delta^n_m\) is nonzero only when \(m = n\). This is vertical fuzziness.

The expansion point \(q\) is the horizontal observable. It represents the location of the detector. If \(p\) is the detector's momentum, the uncertainty relation becomes
\[
\Delta q \cdot \Delta p \geq \hbar/2.
\]
This is horizontal fuzziness.

Hence by replacing the fields with the corresponding Taylor series we get a theory in which both the result of the experiment and its location are fuzzy; QJT implements both vertical and horizontal fuzziness. 

To replace the fields by their Taylor series might sound trivial, and it would be if the expansion point were a fixed point in space. But the last equation means that we expand the fields around a fuzzy point.

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Renormalization

Yesterday we argued that in a Quantum Field Theory (QFT) of gravity cannot exist, due to incompatible assumptions about the observer's mass. Today I will discuss the analogous situation for non-gravitational interactions such as electromagnetism.

An experiment is still an interaction between a system and an observer, and the outcome still depends on the observer's physical properties. Since the predictions of QFT are independent of the observer's charge and mass, we have still made some hidden assumptions, namely

• The observer's charge is zero, so the observer does not disturb the fields.
• The observer's inert mass is infinite.

What makes the situation different from yesterday is that this double limit exists. Only in the case of gravity, where charge equals heavy mass, are charge and inert mass related. For non-gravitational interactions charge and mass are independent, and we can assume both that charge is zero and that mass is infinite.

However, there is a catch. In reality the observer's mass is large but finite, and QFT is only an approximation, valid for systems where all relevant energy scales are much smaller. But this means that all energy scales must be smaller than the observer's mass, including the energy of virtual particles. So the observer's mass essentially becomes a cut-off which limits the energy of virtual particles.

The fundamental theory has horizontal fuzziness. It can be approximated by QFT without horizontal fuzziness, but too energetic virtual particles must be removed. Otherwise some of them may collide with the detector and displace it, making its location fuzzy. Once quantities have been calculated at a given cut-off, we can let the observer's mass approach infinity, making sure that the observed quantities change consistently. This process is called renormalization.

Renormalization is today a standard part of a physicists toolbox, but it was not always uncontroversal. For two decades after the discovery of quantum mechanics, applying it to fields yielded nonsensical infinite results. In the late 1940s people like Schwinger, Feynman and Tomonaga realized how to extract finite results, essentially by subtracting two infinite numbers. Older physicists like Dirac remained skeptical and found such hokus-pokus unsatisfactory, even if the recipe gives extremely accurate results. My perspective is similar to Dirac's. Renormalization is the price to pay for applying a theory without horizontal fuzziness to a situation where horizontal fuzziness is present.

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An Alternative Formulation

Today I will repeat the argument from yesterday using a slightly different formulation. 

Every experiment is an interaction between a system and an observer, and the outcome of the experiment depends on the physical properties of both. In particular, it depends on the observer's mass. Neither general relativity nor Quantum Field Theory (QFT) make predictions that depend on the observer's mass, so some hidden assumptions must have been made. It is clear what those assumptions are:

• In general relativity, the observer's heavy mass is zero, so the observer does not disturb the fields. This observer is essentially what Einstein calls a test particle.
• In QFT, the observer's inert mass is infinite, so the observer knows where he is at all times. In particular, the observer's position and velocity at equal times can be measured simultaneously to arbitrary precision.

Now the equivalence principle asserts that heavy mass equals inert mass, i.e. zero equals infinity. This is the basic problem with a QFT of gravity.

Hence quantum gravity requires that the observer's mass is finite and nonzero. Then we can not ignore the interaction between the observer and the gravitational field, and we can not ignore horizontal fuzziness.

Let us spell out the second assumption in more detail. Let \(q\) be the observer's position and \(p\) his momentum. The accuracy with which they can be simultaneously measured is limited by the uncertainty principle:
\[
\Delta q \cdot \Delta p \geq {\hbar\over2},
\]
where \(\hbar\) is Planck's constant divided by \(2\pi\). The observer's velocity \(v\) is given by \(p = Mv\), where \(M\) is the observer's mass. Hence
\[
\Delta q \cdot \Delta v \geq {\hbar\over2M}.
\]
In order to know the observer's position at all times, we must simultaneously know his position and velocity at equal times without any uncertainty. Hence the right-hand side of the equation above must vanish, i.e. \(\hbar/M = 0\). There are two ways to achieve this:

• \(\hbar = 0\). This is classical field theory including classical gravity.
• \(M = \infty\). This is quantum field theory without gravity.

Hence we can ignore horizontal fuzziness in those two limits, and only then. If Planck's constant is nonzero and the observer's mass is finite, which is the case in every real-world experiment, horizontal fuzziness is always present in principle.

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Horizontal Fuzziness

The difference between classical and quantum physics is fuzziness. In classical mechanics all quantities can be measured to arbitrary precision, but observables in quantum mechanics are subject to quantum fluctuations. There is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known; this is Heisenberg's uncertainty principle. Hence measurements in quantum mechanics are fuzzy.

Our most accurate theories are theories of fields, i.e. systems which depend on space as well as time. Electromagnetism, weak and strong interactions are all described by Quantum Field Theory (QFT), i.e. quantum theory applied to fields. In QFT we have the same kind of fuzziness as in quantum mechanics. The result of an experiment is fuzzy, but the location of the experiment is not. This is unphysical in my opinion. The experiment takes place inside a detector, and in order to know where the detector is located, we must measure its position, e.g. with rulers or a GPS receiver. This measurement is a separate physical experiment and as such subject to quantum fluctuations. Hence we need to replace QFT with a more general theory which takes into account that the detector's location is fuzzy.

A fundamental theory must hence have both vertical fuzziness (the result of an experiment is fuzzy) and horizontal fuzziness (the location of the experiment is also fuzzy). We expect that QFT is recovered in the limit that horizontal fuzziness can be ignored. This is the case when the detector is a classical object whose position at all times can be known without fuzziness. This amounts to a hidden assumption that the detector's inert mass is infinite.

Now it is easy to see why QFT is incompatible with gravity. The equivalence principle asserts that inert mass equals heavy mass. However, an infinitely heavy detector will interact with the gravitational field and immediately collapse into a black hole, which sucks up the rest of the universe. This is not a very good description of most experiments. 

Quantum mechanics reached its final formulation a century ago, and people attempted to apply it to gravity soon thereafter. There has been essentially no progress on this problem over the past century, and the reason is now clear. By ignoring horizontal fuzziness, a hidden assumption about an infinitely massive detector is introduced, and gravity does not work well in the presence of infinitely massive objects. There have also been other well-advertised approaches to a quantum theory of gravity. It is not clear to me to what extent they are related to field theory, but it does not really matter. Absence of horizontal fuzziness amounts to the same hidden assumption about an infinitely massive detector, and hence such theories are also incompatible with gravity.

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Dad's Theory of Physics

A long time ago I aspired to be a theoretical physicist. Eventually I came up with a discovery that I now believe is quite significant, but it took more than a decade before I started to understand what it meant. Since my postdoc only lasted four years, I ran out of funding and went on to do something else. However, my wife insists that I must explain to our children what I spent so much time on even after I had left academia. This will be done in a series of posts over the next three weeks. 

Explaining a discovery in mathematical physics is not so easy, especially not to lay-people. I could say (and have said) that I generalized the Virasoro extension of the algebra of diffeomorphisms on the circle to higher-dimensional manifolds, and discovered how to build off-shell representations thereof. Alas, that sentence is generally met with a blank stare. Instead I will try to explain the main physical ideas, initially using a minimum of formulas. This will be the topic of the first week, where I introduce the notion of horizontal fuzziness and explain why it is a necessary ingredient in any quantum theory of gravity.

In the first week I plan to cover the following topics:

• Horizontal Fuzziness
• An Alternative Formulation
• Renormalization
• Quantum Jet Theory
• Partial and Complete Observables

The second week I will describe the mathematical discovery underlying these ideas. This will require quite a bit of mathematical formalism, but I will try to introduce the necessary background material first. What should be emphasized is that the ideas introduced during the first week do not come out of thin air. Instead, they are the physical interpretation of the off-shell representations of the multi-dimensional Virasoro algebra.

The third week I will discuss some consequences of horizontal fuzziness. Just as vertical fuzziness (quantum mechanics) is different from no fuzziness (classical mechanics), we can expect that horizontal fuzziness is significant too.

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