welcome back to science click today spin at the fundamental scale matter is composed of particles that exist in different versions and which make up everything around us particles have properties a mass or an electric charge for instance which Define their behavior in 1922 the stern gar experiment identified a new property of particles when sending a beam of silver atoms through an inhomogeneous magnetic field it splits in two half the particles are deflected upwards the other half downwards as if there were two types of electrons reacting differently we first imagined that the electrons in the atoms behaved like tiny marbles spinning on themselves thus generating a small magnetic field like an electromagnet this would explain why they are deflected but not why the beam splits in two the electrons would be more or less aligned with the magnetic field and therefore more or less deflected it was later calculated that this explanation would require electrons to rotate faster than light an absurd result today Quantum field Theory our most advanced model describes particles as dimensionless points they have no size and therefore cannot rotate on themselves in short particles seem to have an intrinsic property just like their mass or charge which we call spin in this video we will try to understand the origin of this property and why the Electron Beam splits in two although it is often held to originate from quantum physics we will see that spin rather comes from the geometry of space rotations and group Theory [Music] a group in mathematics is a set of operations which can be combined or reversed for example the set of quarter turn rotations 0° 90° 180° and 270° forms a group a set of operations that can be combined or reversed groups are very useful for describing the symmetries of an object this gemstone for example is symmetric under this group because it does not change appearance after any of these four rotations this next Stone on the other hand is not symmetric since it changes appearance depending on the orientation this particular group made up of four rotations is called by mathematicians Z4 which simply means that it is generated by an operation which cancels itself when applied four times in a row of course this is just one example of a group among many the hours on a clock are described by z12 since this group returns to the starting point after 12 rotations there are also infinite groups like Z squ which contains all displacement of this checkerboard and even continuous groups like that of all rotations between 0 and 360° we call it s 2 because it describes the rotations in two Dimensions the S SO3 group describes the rotations of space in three dimensions and in special relativity we study the group s SO3 comma 1 which describes the rotations of SpaceTime with three dimensions of space and one dimension of time however let's focus on our first example which is simpler and contains only four rotations [Music] to understand how a group can manifest itself and act on objects let's consider these three gemstones and look at how they transform under the action of rotations the rotation group has no effect on the blue stone the red stone on the other hand changes and each of the four operations generates a new unique appearance finally the green stone changes after the first rotation but returns to its initial State as soon as the second rotation occurs the blue stone is always in the same state the red stone can adopt one of four states and the green stone one of two states let's now imagine that these Stones represent particles as physicists we know neither the appearance nor the structure of particles they are fundamental object Elementary building blocks without Dimension however we know that some particles can adopt different states depending on their orientation and even if we do not perceive these different states at our scale we can imagine that they would have some sort of physical effect during experiments let's invent a new abstract space the state space in which we plot each state that a particle can adopt each particle has its own State space let's now rotate the red particle by 90° its state has changed we can say it too has rotated in the abstract space of states for the same 90° rotation the state of the blue particle does not rotate while that of the green particle rotates twice as much the key idea is that we can associate to each rotation in space an abstract rotation in the corresponding State space this construction allows us to distinguish the behavior of the three particles the state of the blue particle rotates zero times that of the red particle rotates one time and that of the green particle rotates two times we call this property the spin number the spin number describes how rotations in physical space are represented in the abstract space the blue particle has a spin number of zero the red particle of one and the green particle of two we say that these are three different representations of the rotation group the spin number is just a label to classify different behaviors when we apply a rotation to an object this bottle has spin one because its state changes as fast as space this button has spin zero because its state doesn't change and this playing card spin two because it comes back to its initial state twice as fast as space to put it simply the spin number of an object measures how fast it comes back to its initial state [Music] in order to make calculations and predictions physicists want to model the world in the language of mathematics instead of buttons bottles or playing cards they look to mathematical tools to describe objects a spin zero object will generally be modeled as a point or a number which has no orientation and always keeps the same state a spin one object will generally be modeled by a vector a sort of Arrow which changes when it is rotated finally a spin 2 object will usually be modeled by a Rank 2 tensor which can roughly be thought of as a double vector and which returns to its state after only 180° so far we have focused on the group zed4 in reality physicists are interested in all possible rotations of SpaceTime still the idea Remains the Same if we know the various representations ways in which rotations can affect an object we have a catalog of mathematical tools to model the universe this is precisely how physicists have described particles some particles have spin number zero like the hix boson its state doesn't change and we therefore describe it by a field of numbers num others have spin number one like the photon which constitutes light light has a directionality called its polarization which only returns to its initial Direction after 360° it is this polarization which allows polarized lenses to block certain light depending on its orientation light is described by a vector field the electromagnetic field finally the graviton the still hypothetical particle of gravity would have a spin number to because in general relativity we describe Gravity by a field of rank two tensors which returned to their state after only a 180° rotation this Spin 2 property is found in the very structure of gravitational waves which deform space in two perpendicular directions with a symmetry of 180° in short the spin number indicates which mathematical tools can be used to describe particles according to how they react to rotations but what about electrons and more generally fermion the particles of matter we often hear that these particles have neither spin zero 1 nor two but 1/2 how can something have spin one half [Music] if we imagine a new gemstone with a spin number one half this means that each rotation would generate only half a rotation in its state space it would thus require not one but two full turns to regain its initial state of course this seems absurd in classical physics all objects must remain the same after a 360° rotation still these spin half particles do exist and even make up within atoms all matter that surrounds us to understand we will have to delve into quantum mechanics the core of quantum mechanics is the superposition principle to understand it clearly let's make an analogy with a coin in classical physics a coin only has two possible States heads or tails however in quantum mechanics as long as we don't observe it the coin can actually be in a Quantum super position a state involving both heads and tails with probabilities it is only when we observe the coin that the universe will have to collapse at random between heads or tails according to these probabilities of course this is just an analogy this phenomenon does not occur for a macroscopic coin but it does occur at the scale of quantum particles to model Quantum superpositions we once again use an abstract space in which heads and tails are just two axes two directions each superposition is a sum which involves heads and tails in arbitrary proportions this state for instance corresponds to the sum heads plus Tails this one to the sum two heads minus three Tails heads and tails Act as coordinates in this abstract space and all superpositions are potential states that the coin could adopt before it is observed when we observe it the probability of getting either heads or tails depends only on the direction of the coin State the more it is aligned with the Tails axis the more probably will'll get tails and vice versa the one thing that's worth noting is that the probabilities of getting either head or tails are the same whether we consider one state or its opposite whether the state is directed in One Direction or the other has no impact its alignment with the heads and tails axes Remains the Same and the probabilities for these results are therefore the same the two opposite States actually represent the same reality and it is impossible to differentiate them with any experiment the state plus heads for instance is physically equivalent to the State minus heads it is this subtlety that will finally allow us to understand spin one half let's return to our example one last time we saw that an object with spin number one/ half was impossible in classical physics because a full turn would bring it to the opposite of its initial state which is absurd however if the object is quantum this is not a problem anymore precisely because the opposite of a Quantum state is physically equivalent to it although mathematically different the idea of a spin half object is therefore allowed within quantum physics we call the mathematical tool that models it a spinner it is an abstract entity which requires two rotations of space to return to its starting point in the state space this is allowed because after one full turn or although it is now in its opposite State the spinner physically describes the same initial situation we might think that spin half objects aren't very interesting given that they still come back to the same physical state after a full turn however there is actually a difference between Spinners and the other tools we've seen so far if we consider one state and its opposite these are physically equivalent but they become different in a super position with another State and if we imagine a field of spinners in which waves propagate the superposition of two waves will be different if we replace one of the two Waves by its opposite Spinners therefore do have interesting properties that differentiate them from other [Music] objects with all these Notions we can finally understand the spin of an electron electrons have spin number 1/2 this means that they can be modeled by a field of spinners which return to their initial state after only two complete rotations if we pick an electron at random it will have a state pointing in One Direction of its abstract State space we call this its spin State physicists have named the axes of this space up and down because electrons in these states are deflected upwards or downwards in the stern experiment if we rotate the electron its spin state will also rotate in its abstract space but only half as fast because its spin number is 1/2 a spin up electron becomes spin down after 180° after a full turn the spin up electron has reached the opposite State minus up but this is physically equivalent to the initial spin up State these two states are analogous to the heads and tails axes of the quantum coin and during a measurement the electron will collapse at random onto one of these two results either up or down it is this property that the experiment highlighted back in 1922 understanding why such particles behave as if they were spinning on themselves is quite Technical and beyond the scope of this video it boils down to the fact that angular momentum which measures how much an object is spinning can be calculated by looking at how an object State changes after a small rotation and since the spin number characterizes how fast the particle State changes during a rotation it therefore acts as a sort of angular momentum internal to the particle and generate similar effects to a spinning motion it is crucial to understand that the spin state of an electron up down or a superposition of the two is not a direction in space but an abstract Direction in a mathematical space invented to describe the particle in fact so far we have visualized rotations in two Dimensions but our space actually has three dimensions in which we can change orientation the abstract space which describes the spin of an electron however only has two Dimensions up and down they are two completely different spaces linked only by the fact that each rotation in one is mapped to a rotation in the other to be perfectly exact quantum mechanics does not use real numbers but complex numbers to describe Quantum states which makes this space harder to visualize let's avoid getting lost in details and summarize everything we have seen one last time particles have an intrinsic property called the spin number the spin number characterizes how fast the state of a particle changes when we rotate it in space some particles have integer spin numbers 0 1 or two for example but quantum mechanics also allows the existence of particles with half integer spin in particular 1 half which need two rotations to regain their initial State when we measure the state of such a particle it randomly collapses onto one of the two axes of its abstract space explaining why the beam splits in two in the Stak experiment finally a few years later in 1928 Bak became interested in the behavior of spinners when we add the dimension of time in the context of Relativity when we change our reference frame which amounts to rotating SpaceTime a spinner rotates according ly within its abstract space however it turns out there are two possible behaviors it can react either in One Direction or the other mathematically a spinner somewhat behaves like a square root there are two of them one positive and one negative in the same way there are two different types of spinners reacting in opposite ways in the direction of time they correspond to particles and antiparticles in this way D mathematically predicted the existence of antimatter [Music] [Music]