What are the automorphisms on the strucuture consisting of the nonzero vectors of a Hilbert space with the orthogonality relation? Let $V$ be an infinite-dimensional complex Hilbert space.   With this space we can associate a relational structure $V^+ = (V^+, \bot)$, where $V^+$ is the set of non-zero vectors in $V$, and $\bot$ is the orthogonality relation.  Every automorphism of $V$ induces another on $V^+$.   What  are other  automorphisms of $V^+$?  This is also a reference-request question.
Motivation: In quantum logic, an abstract approach to quantum theory, one examines (structures arising from) $V^+$ instead of $V$ itself.
First observations:  For an automoprhism $\phi$ of $V^+$, the image along  $\phi$ of a line through the origin must be contained in another.  The same thing is true of planes through $0$.  This reminiscent of collineations in projective geometry.  But it's not clear to me if an automorphism on $V+$ has to preserve additions.  It certainly does needs to preserve an addition of two orthogonal elements, however.
 A: Let $P$ be the set of lines through the origin in $V$ (i.e., the projective space of $V$).  Given any automorphism $\phi$ of $(P,\perp)$, we can obtain an automorphism of $V^+$ by arbitrarily picking bijections $l\setminus\{0\}\to\phi(l)\setminus\{0\}$ for each $l\in P$, and every automorphism of $V^+$ arises in this way.  So, we will instead consider automorphisms of $(P,\perp)$.
Note now that any automorphism of $(P,\perp)$ is a collineation of the projective space $P$: given distinct $m,n\in P$, $l$ is on the line between $m$ and $n$ iff $l\subseteq m+n$ as subsets of $V$, which holds iff every line orthogonal to both $m$ and $n$ is also orthogonal to $l$.  By the fundamental theorem of projective geometry, this implies any automorphism $\phi$ of $(P,\perp)$ is induced by a semilinear automorphism $f$ of $V$; that is, a bijection $f:V\to V$ which preserves addition and for which there exists a field automorphism $\sigma$ of $\mathbb{C}$ such that $f(cv)=\sigma(c)f(v)$ for all $c\in\mathbb{C},v\in V$.
So, it suffices to determine which semilinear automorphisms $f$ of $V$ preserve orthogonality.  Supposing we have such an $f$ (with associated automorphism $\sigma$ of $\mathbb{C}$), let $v,w\in V$ be nonzero orthogonal vectors of the same norm.  Then for any $r\in\mathbb{R}$, $v+rw\perp-rv+w$, and thus $f(v)+\sigma(r)f(w)\perp -\sigma(r)f(v)+f(w)$.  In the case $r=0$, this tells us $f(v)$ and $f(w)$ are orthogonal.  In the case $r=1$, it then tells us $f(v)$ and $f(w)$ have the same norm.  For general $r$, we then conclude that $\sigma(r)$ is real.  Thus $\sigma$ restricts to an automorphism of $\mathbb{R}$, which must be the identity since $\mathbb{R}$ has no nontrivial automorphisms.  We thus conclude that $\sigma$ must be either the identity or complex conjugation.
The discussion above also shows that $f$ preserves when two orthogonal vectors have the same norm; it follows that $f$ preserves when two vectors have the same norm (take a third vector of the same norm orthogonal to both of them).  Multiplying $f$ by a real scalar (we only care about the induced map on $P$ anyways), we may assume $f$ maps unit vectors to unit vectors, and thus $f$ preserves the norm since $f$ is $\mathbb{R}$-linear.  When $\sigma$ is the identity, this means $f$ is a unitary automorphism of $V$.  When $\sigma$ is complex conjugation, the means $f$ is an antiunitary automorphism of $V$, i.e. a conjugate-linear automorphism that satisfies $\langle f(v),f(w)\rangle=\overline{\langle v,w\rangle}$.  (If you identify $V$ with $\ell^2$, say, an antiunitary automorphism is just a unitary automorphism composed with the map that conjugates each entry of a sequence.)
Since any unitary or antiunitary automorphism of $V$ obviously preserves orthogonality, this gives us a complete characterization: the automorphisms of $(P,\perp)$ are exactly those maps which are induced by unitary or antiunitary automorphisms of $V$.
(All of this works for any complex inner product space $V$ of dimension at least $3$.)
