I need an introduction to Floer theory that's suitable for perhaps a beginning math grad student or a 2nd year physics grad student. The wiki article is sufficiently over my head that it reads as "bar" "bar" "bar". Editor, please improve the tags.
The reason this comes up is that an answer to a problem on unitary matrices and Hilbert space bases was answered with a reference to Floer theory.
On
The reason I had asked this question is because a Floer theory proof was needed for a physics paper I was writing and I didn't want to include a proof I didn't understand. Eventually I gave up on the hope that someone would explain it to me and began reading the mathematics literature.
Those looking for a translation of the mathematics jargon into physics jargon (of the sort that every physics grad student is taught) might look at the paper I'm planning on submitting to Jour. Math. Phys., or perhaps Phys. Rev X. Section III, "Hamilton's Equations" is a translation of a short Floer theory proof that Sam Lisi gave in response to the question "Given two basis sets for a finite Hilbert space, does an unbiased vector exist?" For completeness, here's the current version:
An Hermitian matrix generates a 1-parameter subgroup of unitary matrices and any unitary matrix is an element of such a 1-parameter subgroup. Since we are translating the problem into classical mechanics we will use $t$ for the parameter. Thus:
$$U(t) = \exp(it\;H).$$
and the unitary matrix of interest is given by $U(1)$. Given an initial state $\vec{v}(0)$, the state at time $t$ is defined by a set of coupled ordinary differential equations:
$$\vec{v}(t) = U(t)\vec{v}(0) = \exp(it\;H)\vec{v}(0)$$
The 1-parameter subgroup (and therefore the unitary matrix) are fully defined by the relationship between $\vec{v}$ and $\dot{\vec{v}}$. In components:
$$\dot{v}_j = i\Sigma_k H_{jk}\;v_k.$$
Replace the complex variables with real and imaginary parts:
$$v_k = p_k + iq_k$$
$$H_{jk} = r_{jk}+is_{jk}$$
If these are compatible with a Hamiltonian $\mathbf{H}$, we have Hamilton's equations:
$$\dot{q}_j = +\partial \mathbf{H}/\partial p_j = \Sigma_k(+r_{jk}p_k-s_{jk}q_k),$$
$$\dot{p}_j = -\partial \mathbf{H}/\partial q_j = \Sigma_k(-r_{jk}q_k-s_{jk}p_k).$$
Compatibility requires that $s_{jk}=-s_{kj}$, which is true since $H$ is Hermitian. Integrating gives the Hamiltonian as:
$$\mathbf{H} = \Sigma_{j\neq k}(r_{jk}(p_jp_k+q_jq_k)+s_{jk}p_kq_j)+\Sigma_{j}r_{jj}(p_j^2+q_j^2)/2.$$
This Hamiltonian, integrated for a period of time $t$, gives the unitary transformation $U(t)=\exp(iHt)$. Note that $\mathbf{H}$ is quadratic in momentum and position and so is a generalization of an harmonic oscillator.
Mathematicians study these Hamiltonians under the label "symplectic geometry." Here we give a brief and rough introduction to the mathematical language. Let $\{\hat{e}_j\}$ be a basis for the positions as a vector space. That is, given a position $\vec{q}=(q_1,q_2,...q_n)$, we treat the sum $\Sigma_j\hat{e}_jq_j$ as an element of a vector space. Similarly, let $\{\hat{f}_k\}$ be a basis for the momenta also with $n$ elements. Combining the two basis sets gives a basis for a $2n$-dimensional vector space $M$. Now define a bilinear map $\Omega$ on $M$ which acts on the basis sets as follows: $$\Omega(\hat{e}_j,\hat{e}_k)=\Omega(\hat{f}_j,\hat{f}_k)=0,$$ $$\Omega(\hat{e}_j,\hat{f}_k)=-\Omega(\hat{f}_k,\hat{e}_j)=\delta{jk}.$$ Without $\Omega$, $M$ is the usual "phase space" of the physicists but the mathematicians prefer to call the combination a "symplectic vector space."
The map $\Omega$ can be thought of as a way of associating positions with momenta. That is, given two elements $u,v$ of $M$ with $\Omega(u,v)=1$, we can think of $u$ as a position and $v$ as its associated momentum. For example, if $\Omega(q_1,p_1)=1,$ then $\Omega(p_1,q_1)=-1$ so $\Omega(p_1,-q_1)=1$. Thus we can think of $p_1$ as a position and $-q_1$ as its associated momentum. This use follows the sense of the usual canonical (or contact) transformations familiar to classical mechanics. This example is one that is typically given in textbooks on the subject; we can swap a position for its associated momentum provided we introduce a minus sign.
Classical mechanics is about the movement of systems through phase space. Suppose a system begins at some particular position. A question of interest is "can the system return to that position at time $t$?" To answer this question, we consider a fixed position with all possible momenta. But Hamilton's equations can be transformed in ways that mix position and momentum. So to understand these questions we need a definition of "initial position" that allows for any possible transformation of Hamilton's equations.
If phase space is not transformed, then the appropriate elements of $M$ to consider are those with particular position and any momentum. This is easy to define by the $\hat{e}_j,\hat{f}_k$ basis elements; we let momentum be in the subspace spanned by the $\hat{f}_k$. Such a subspace has dimension $n$, just half that of $M$. More generally, consider the momentum subspace resulting from any canonical transformation along with a specification of position. Such a subset of $M$ defines an initial value problem in classical mechanics; the mathematicians call such a subset a "Lagrangian submanifold".
We now consider the canonical transformation from $q_j,p_j$ to $\rho_j,\sigma_j$ generated by: $$F = \left(q_j\sqrt{\rho_j^2-q_j^2}+\rho_j^2\sin^{-1}(q_j/\rho_j)\right)/2.$$ This gives $p_j$ and $\sigma_j$ as: $$p_j = \partial F/\partial q_j = \sqrt{\rho_j^2-q_j^2},$$ $$-\sigma_j = \partial F/\partial \rho_j = \rho_j\sin^{-1}(q_j/\rho_j).$$ Solving for $p_j$ and $q_j$ in terms of $\sigma_j$ and $\rho_j$ we have: $$p_j=\rho_j\cos(\sigma_j/\rho_j),$$ $$q_j=\rho_j\sin(\sigma_j/\rho_j).$$ Putting $\rho_j=1$ in the new coordinates defines a Lagrangian submanifold of $M$ for which $\rho_j^2=p_j^2+q_j^2=1.$ And this subset of phase space corresponds to the vectors of phases in Hilbert space. The new momentum consists of a product of $n$ copies of complex phases so it can be called a torus; since it is also Lagrangian, it is a "Lagrangian torus". The torus as we've defined it has a phase freedom. That is, if we add the same phase $\alpha$ to all the $\sigma_j$, the result will be a new vector that is also a vector of phases and that represents the same quantum state. This is just the usual arbitrary complex phase present in a quantum state vector. To eliminate it, the mathematicians prefer to identify equivalent vectors and so work with the equivalent torus in $CP^{n-1}$.
Cheol-Hyun Cho3 refers to our $CP^{n-1}$ torus as a "Clifford torus", an extension of the usual definition. His paper is perhaps the first proof that a Hamiltonian flow cannot "displace" such a torus, that is, move it in such a way that it no longer intersects with itself. Other papers that prove the existence of the intersection are [4,5] and it can be deduced from [6-8]. This completes the proof that an unbiased state exists for two bases. In addition, computer calculation with random unitary matrices failed to find any counter examples and Philip Gibbs9 proved the $n=3$ case in 2009.
3 C.-H. Cho, “Holomorphic discs, spin structures, and floer cohomology of the Clifford torus,” Int. Math. Res. Not. 35, 1803–1843 (2004), math / 0308224.
4 P. Biran and O. Cornea, “Lagrangian quantum homology,” The Yashafest, Stanford (2007),
math.SG / 0808.3989.
5 P. Biran and O. Cornea, “Rigidity and uniruling for lagrangian submanifolds,” Geom. Topol. 13, 28812989 (2009), math.SG / 0808.2440.
6 C.-H. Cho and Y.-G. Oh, “Floer cohomology and disc instantons of lagrangian torus fibers in fano toric manifolds,” Asian J. Math. 10, 773–814 (2006), math / 0308225.
7 M. Entov and L. Polterovich, “Quasi-states and symplectic intersections,” Eur. Math. Soc. 81, 75–99 (2006), math / 0410338.
8 K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, “Lagrangian floer theory on compact toric manifolds: survey,” (2010), math.SG / 1011.4044.
9 P. Gibbs, “3x3 unitary to magic matrix transformations,” (2009), vixra 0907.0002.
To my knowledge, the best graduate level introduction to Floer theory is this (new) book:
Michèle Audin et Mihai Damian, Théorie de Morse et homologie de Floer
I hope the linguistic barrier isn't too big an obstacle but I don't think an introduction this wide and with this level of details exists in any other language. (I'm quite confident it will be translated, but when?).