Information loss, and emergent type III1 algebras in time

<!--HTML--><p><span style="color:rgb(0,0,0);"><span style="-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);display:inline !important;float:none;font-family:Helvetica;font-size:14.666667px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">I discuss a key example to demonstrate that&nbsp;the decay of the two-point function (clustering in time)&nbsp; in a black hole background holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity.&nbsp; In the thermodynamic limit of infinite entropy (infinite volume or large N), the operators that cluster in time are expected to form an algebra. I show that this algebra is a unique and very special infinite dimensional algebra called the&nbsp;</span><span style="-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);font-family:Helvetica;font-size:14.666667px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">III_1&nbsp;</span><span style="-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);font-family:Helvetica;font-size:11pt;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">factor. I&nbsp;prove a generalization of a conjecture of Leutheusser and Liu to arbitrary out-of-equilibrium states. I explicitly construct the C-algebra and von Neumann subalgebras associated with time bands and more generally, arbitrary open sets of the bulk spacetime in the strict N\to\infty&nbsp;limit. The emergence of time algebras is intimately tied to the second law of thermodynamics and the emergence of an arrow of time.</span></span></p>