## Rethinking Unitarity: Embracing Isometry in an Expanding Universe

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Unitarity, as the principle is called, says that something always happens. When particles interact, the probability of all possible outcomes must sum to 100%... But what once seemed an essential scaffold may have become a stifling straitjacket preventing physicists from reconciling quantum mechanics and gravity. A century ago, the pioneers of quantum mechanics made a surprising discovery – one that elevated unitarity from common sense to a hallowed principle. Mathematically, the quantum world operates by more complicated numbers known as amplitudes. An amplitude is essentially the degree to which a particle is in a certain state; it can be a positive, negative or imaginary number. To calculate the probability of actually observing a particle in a certain state, physicists square the amplitude which gets rid of the imaginary and negative bits and produces a positive probability. Unitarity says the sum of these probabilities must equal 1… As a particle’s state changes, its amplitudes change too. In working out how a particle is allowed to evolve or interact, physicists use the fact that amplitudes never change in a way that disrupts the fixed sum of their squares.

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Physicists keep probabilities and amplitudes in line by tracking how the quantum state of a particle moves around in Hilbert space – an abstract space representing all possible states available to the particle. The particle’s amplitudes correspond to its coordinates in Hilbert space… Unitarity dictates that a physically allowed change must correspond to a special “unitary” matrix that rotates the particle’s state in Hilbert space such that the sum of the squares of its coordinates equals 1.

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And yet, this bedrock assumption seems to conflict with the universe that surrounds us. While our expanding universe is a perfectly valid solution to the equations of general relativity, physicists have increasingly realized that its growth spells trouble for quantum mechanics, by presenting particles with an expanding smorgasbord of options for where to be and how to behave. As space grows, how can the Hilbert space of possibilities not grow with it?

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Andrew Strominger and Jordan Cotler of Harvard University argue that a more relaxed principle called isometry can accommodate an expanding universe while still satisfying the stringent requirements [of] unitarity. In a paper earlier this year, the two homed in on a class of transformations known as isometries. An isometric change resembles a unitary one with added flexibility. Think of an electron that can occupy two possible locations. Its Hilbert space consists of all possible combinations of amplitudes in the two locations. These possibilities can be imagined as the points on a circle –– every point has some value in both the horizontal and vertical directions. Unitary changes rotate states around the circle but do not expand or shrink the set of possibilities.

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To visualize an isometric change, though, let the universe of this electron swell just enough to allow a third position. The electron’s Hilbert space grows, but in a special way: It gains another dimension. The circle becomes a sphere, on which the particle’s quantum state can swivel around to accommodate mixtures of all three locations. The distance between any two states on the circle holds steady under the change –– another requirement of unitarity. In short, the options increase, but without unphysical consequences…

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