# What is the complete set of parameters that describes a photon in free 3D space?

I'm trying to understand what parameters uniquely define the state of a photon, and I can't seem to find a straightforward reference anywhere. Wikipedia is either unclear here, or over my head.

If you were to restrict yourself to parameters that are integer-valued or real-valued (if this is possible), how many parameters would you need to describe a photon from a fixed reference point?

The best I can come up with is at least 6, but I'm unsure about some of this. Here's what I have:

• 3 real-valued parameters to describe its x,y,z direction in space.
• 1 real-valued parameter to describe its frequency, or equivalently, its energy.
• 1 real-valued parameter to describe its phase offset.
• 1 integer-valued parameter to describe its spin angular momentum, which is either +1 or -1 times ħ.

There are two more parameters that I'm unsure about.

First, Wikipedia says "each photon carries two distinct and independent forms of angular momentum" and lists spin angular momentum (SAM) and orbital angular momentum (OAM). SAM seems to pretty clearly be a property of a single photon, however, the language used in the Photon article and related articles when discussing OAM makes it sound like OAM might be a property of a set of photons, and not just of a single photon. Is it a property of a single photon?

Second, photons are often depicted as a magnetic field and an electric field oriented 90 degrees from each other, perpendicular to the direction of travel of a photon, such as this image:

Can that field have any orientation relative to some fixed reference? Can I have two photons where one's fields are perfectly aligned with my frame of reference, but the other is rotated?

Depending on the answers to the above, the tally comes to 8 parameters:

• The integer-valued parameter to describe its orbital angular momentum.
• The real-valued orientation (rotation) of the photon's fields relative to the reference point.

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My apologies if the question itself contains mistruths or misunderstandings. My background is in computer science, and I work for an RF engineering company which is where my curiosity stems from.

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Edit: After reading through the link provided by Emilio Pisanty, it appears that there are two forms of orbital angular momentum - external and internal, where internal refers to the orbital angular momentum of a single photon. Thanks Emilio!

• Aug 10 '20 at 21:57