Does God play dice and when A short intro to the measurement problem in Quantum Mechanics The problem sort of Quantum systems can be in superpositions seem to have several values for an observable at once If we try to measure systems must decide collapses into corresponding basis state But when does the measurement happen what determines whether system has been measured or not This way or that Send electrons through double slit get interference pattern not additive pattern Act like waves cannot go through just one slit or other to get interference Measure which slit interference pattern disappears Get additive pattern each electron decides and goes through just one of the slits Weird but electrons are small To meow or not to meow E Schr dinger 1935 cat in a box with radioactive element and flask of HCN Geiger counter linked to hammer breaks flask if atom decays Evolves into superposition cat does not live or die until measured When is the cat measured When box is opened Does it measure itself Down to size At some point superpositions break down give us sharp classical world Several proposals for how Conscious observers are special seriously Maybe QM breaks down at some size scale but what scale and what is meant by size Maybe universe branches into parallel universes at each measurement we live in one particular branch ok but how do we ever know Whoa so what was the problem again Double slit revisited Distribution when slit 1 is open when slit 2 is open Classically we expect when both slits are open by addition of probabilities Does not agree with reality But why do we expect addition of probabilities Probability axioms Classical probability theory can be constructed from the following axioms 1 P A 0 for all events A 2 P A and B P A B P B 3 P A or B P A P B for mutually exclusive events A and B 4 P A 1 if A occurs with certainty But why The Dutch Book argument It s all about money A bookie lets you bet arbitrary amounts of money on a set of events and all possible combinations Bookie assigns a probability p any number he wishes to every event and combination of events If an event or combination does not occur you lose the money you bet on it If it does occur bookie pays you 1 p times your bet The Dutch Book argument If probabilities are assigned in a way which violates the usual probability axioms you can find a combination of bets such that the bookie runs a net loss every single time regardless of what events occur If probabilities are a measure of belief risk then probability axioms are rules which all agents must follow to avoid guaranteed loss But this assumes that you can bet on any combination of events and crucially that you can verify whether any event occurred or not Dutch and double slit Interference pattern emerges only if it is impossible even in principle to infer which slit the particle went through There must not exist any which way information anywhere in the universe Cannot check whether particle went through slit 1 or 2 Are not events Dutch Book does not apply no addition of probabilities If we can infer in principle which way the particle went then Dutch book does apply Particle must obey classical probability theory and we get additive pattern no interference Has effectively been measured Measurements dice and determinism So an observable is measured when the environment records information about its value Key is interactions with the environment no need for conscious observers But when does environment collapse Just pushing the problem outwards here Does environment require conscious observers to collapse And what about conscious observers themselves Can they be in superposition states Don t they measure themselves Information vs observers What if we send a conscious observer through a double slit Does the observer know which slit he she went through If so must see slit i e absorb photons This leaves a record in the environment Alternatively observer absorbs no photons but then does not know which slit he she went through No which way information not even in brain of observer hence there can be an interference pattern Not just disorientation Issue is not just that the observer does not know where he she is As long as there does not exist any information about path of observer we can obtain results inconsistent with assuming that the observer went along a definite path at all Point is what information is in principle available not whether any conscious observer observes it or not Of cats and curiosity But what of Schr dinger s poor kitty It lives or dies depending on which path it takes so it has to know Cat knows what happens to it and the inside of the box so it does not describe itself with a superposition state If no information leaks out we cannot know even in principle so we do describe the cat and nucleus as being in a superposition state No contradiction here Cat sees itself living or dying with 50 probability we see the same thing if we open the box Of cats and curiosity No problem as long as we think of quantum states as being statistical functions assigned to systems by observers and depending on what information is in principle accessible to an observer But wait a minute cat says it s dead or alive with 50 probability we say it s in an equal superposition state Surely those aren t equivalent statements A cat with nine unitary lives Sure aint Superposition state has the form alive e Knowing phase we can apply a unitary i dead transformation use interference to e g make cat dead or alive with certainty sort of a Catamard gate But no record must allow us to infer whether cat was dead or alive along the way Cat s memories must be wiped In the end cat s and our state assignment agree and neither knows that they ever differed Fine fine But how can physical quantities have several values at once or no definite value at all How can a cat be dead and forget The Matrix strikes again How to understand what it means that position momentum etc is not defined without abandoning Common Sense and Sanity Think of a computer program trying to explore the computer it is running on that d be us Can discover stuff only by calling functions provided by the OS float FindElectronPosition float FindElectronMomentum Gets a value whenever it calls a function But what value does a function have when the program is not calling it Does this question even make sense The Matrix strikes again Can make sense e g if the function looks like this float FindElectronPosition global